
Class LJ3 /OCo I 



Copyright In . 



COPYRIGHT OEPOSm 



Improvement and the Distribution 
of Practice 

By 

Robert Alexander Cummins 

B.S. (ILL. WES.), M.A. (ILL.), PH.D. (COL.) 

Head of Department of Education, Simpson College, 
Indianola, Iowa 



Teachers College, Columbia University 
Contributions to Education, No. 97 



Published by 

JEcaclieiG College, Columbia JHniuersitp 

NEW YORK CITY 

1919 



jfionograpn 



Copyright, 1919, by Robert Alexander Cummins 



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©CLA535038 



ACKNOWLEDGMENTS 

The author wishes to express his sincere appreciation 
to the superintendents and principals in whose schools 
the experiments described in this report were carried 
out and to the teachers who rendered valuable assis- 
tance in conducting part of the practice. 

His greatest obligation is to Dr. E. L. Thorndike, 
whose keen criticism and wise council have given to 
this study whatever of merit it may possess. Acknowl- 
edgment is also due Dr. R. S. Woodworth and Dr. J. 
McKeen Cattell for helpful criticisms received around 
the seminar table and Dr. Monroe, Dr. Suzzallo, Dr. 
Strayer and Dr. Snedden, in whose classes many help- 
ful and inspiring suggestions bearing on the study were 
received. Much help in the treatment of the data was 
rendered by Dr. A. I. Gates and Dr. W. A. McCall. 

Great credit is due his wife for help of another kind, 
for it was only through her constant sacrifice and devo- 
tion during the years of necessary separation that such 

accomplishment was made possible. 

R. A. C. 



CONTENTS 

PAGE 

Introduction I 

Section I. Learning French Vocabularies by Adults ... .3 

Nature of the Experiment 3 

Results of the Experiment 5 

Discussion of the Two Sorts of Distribution 7 

Section II. The Lyndhurst Experiments with Grade Pupils 11 

Description 11 

Character of the School Population 1 1 

Subjects used in the Experiment 1 1 

Content-material used 12 

Plan of the Experiment 15 

Administration of the Experiment J 8 

Scoring and Tabulating the Results 20 

Results of the Lyndhurst Experiments 23 

Tables of Frequencies 23 

Comparison of the Two Sorts of Distribution 29 

Geography and History 29 

Single-column Addition 34 

Short Division 38 

Section III. The Ohio Experiments with Grade Pupils . . 41 

Description 41 

Character of the School Population 41 

Subjects used in the Experiment 41 

Content-material used 42 

Plan of the Experiment 42 

Administration of the Experiment 45 

Scoring and Tabulating the Results 46 

Results of the Experiments 46 

Tables of Frequencies 46 

Factors Conditioning Improvement 5 1 

Section IV. The Experiment with Normal School Students 55 

Nature of the Experiment 55 



vi Contents 

PAGE 

Results of the Experiment 57 

Discussion of the Results 58 

Section V. Summary of Results of all the Experiments . . 62 

Section VI. The Amount of Improvement and its Relation to 

Initial Ability . 66 

Amount of Gross Gain 66 

Relation of Initial Score to Gross Gain 68 



Improvement and the Distribution of Practice 

INTRODUCTION 

It is the purpose of this monograph to report measurements of 
learning, especially by school children under ordinary classroom 
conditions, and with special reference to the distribution of time 
given to practice. The particular comparison made is between 
(i) a series of practice-periods approximately equal in length, and 
(2) a series of practice-periods decreasing progressively in length. 

Some results obtained by Ebbinghaus ('85, pp. no ff.) and Jost's 
('97) conclusions that "in the case of two associations of equal 
strength, but of different ages, a new repetition has a greater value 
for the older of the two, and that the older association weakens less 
with disuse in a given time," suggest that the second form of dis- 
tribution may be the more favorable in the case of forming serial 
connections with senseless material. General theory also suggests 
that such a distribution might prevent wasteful overlearning in 
both initial practice and reviews in the case of a group of bonds 
which form a limited unit of instruction, as, for example, learning 
a new word in spelling, a vocabulary in a foreign language, or the 
anatomy of an animal. Whether such a distribution would prove 
essentially more favorable in such miscellaneous work as improving 
penmanship, oral speech, computations of various kinds, and the 
like, is more problematical. 

The experiments to be reported here concern : 

A. The learning of French vocabularies by adults. 

B. The learning of geographical and historical facts by school 
children, after the following manner: A brief condensed summary 
of the facts was studied. The knowledge thus gained was measured 
by giving the summary, with certain important words omitted, as a 
completion test. 

C. Improvement in single-column addition by school children and 
by adults. 

D. Improvement in short division by school children. 



2 Introduction 

There is thus a range from the definite formation of new bonds 
from zero strength as in A, to the vague general improvement of a 
function already fairly well established, as in C and D. 

Besides the comparison of the two sorts of distribution of time the 
experiments with school children furnish valuable data concerning 
the use of the practice method as a means of instruction, and con- 
cerning the relation of initial achievement to improvement. 



SECTION I 
LEARNING FRENCH VOCABULARIES BY ADULTS 

The first study presented, though the last one completed, has to 
do with the formation of new bonds from zero strength, i. e., the 
learning of a relatively new thing, as, for example, a new word 
or object. 

THE NATURE OF THE EXPERIMENT 

The experiment reported in this chapter was carried out in the fall 
of 1917, with five graduate students in Teachers College acting as 
subjects. The group was composed of two women and three men, 
including the author himself. Three of the subjects (2, 4, and 5) 
were beginning students in French, and the other two (1 and 3) had 
studied some French several years ago. These facts will be given 
due consideration later. 

The material learned consisted of French words which were pre- 
pared as follows : first, 1 ,200 ordinary French words were selected 
and written, together with their English equivalents, on separate 
slips of paper. These slips were placed in a box and shuffled. The 
words were then drawn out by chance, arranged in groups of ten, and 
rewritten. From the entire list thus formed two smaller lists of 150 
words each were selected by chance. We first eliminated, however, 
all those words which were the same in English, or which were sug- 
gestive of the English meaning. Each list of 150 words was sub- 
divided into five groups, since by trial it was found that approxi- 
mately thirty French-English words could be read over in one 
minute, reading at a convenient rate. Thus it would require just 
five minutes to read over once an entire list of 150 words. 

The following set of instructions, a copy of which was given to 
each of the subjects, will make clear the conditions of the experiment. 

EXPERIMENT IN LEARNING 

FRENCH VOCABULARIES 

The purpose of the experiment is to compare two different distributions of 
time in the learning of the vocabulary of a foreign language: A — a schedule 



4 Improvement and the Distribution of Practice 

in which the study- periods are of equal length, and B — a schedule in which 
the study-periods decrease progressively in length. 

DIRECTIONS 

i. Be sure you understand all the following directions before beginning the 
learning of the words. 

2. Select a time of the day that is least likely to be interrupted for the next 

sixteen days, and use the same time of day throughout. 

3. Note that there are two separate lists, each containing 150 French-English 

words, and each having its own time-schedule. 

4. It should require just five minutes to read over one entire list of 150 

French-English words, i. e., at the rate of thirty words per minute. 

5. When ready to begin, place a timepiece, having a seconds hand, within 

easy view. Begin, when the seconds hand is at zero, and read over the 
French-English words, glancing at the timepiece at the end of every 
group of thirty words, in order to regulate the rate of reading so as to 
cover just thirty words per minute. 

6. Articulate the words under the breath, pronouncing the French words as 

best you can without undue waste of time. 

7. At the end of each five minutes (which should also be at the end of the 

list of 150 words, if you have properly timed the rate), allow a rest of 
one minute, then proceed to go over the entire list again, and so on, 
until the required amount of time has been put in for that practice- 
period according to the time-schedule. 

8. Having put in the required amount of time with list No. 1, allow two or 

three minutes' intermission, and then proceed with list No. 2, if the 
time-schedule calls for any practice on that day, following the same 
general instructions. 

9. In case it should be found absolutely necessary to deviate from the above 

directions, always do the next best thing possible, making note of such 
deviation on the blank provided. 
10. Come to room 300 T. C. on the day after the experiment is finished for 
the purpose of taking the test. 
All necessary materials for the test will be provided. 

Note. Keep a daily record as called for on the blank provided. 

The daily blank mentioned on the instruction sheet provided for 
a record of the hour when the study was done, the general condi- 
tions, i. e., physical, mental, etc., whether poor, good, or excellent, 
and whether or not there were any interruptions or other irregu- 
larities in the work. 



Learning French Vocabularies by Adults 5 

Table 1 gives the exact date of each practice, the amount of time 
spent, and the number of repetitions of the bonds. 

TABLE I 



Date of Each 


Equal Schedule 


Reducing Schedule 


Study- Period 


Time Spent 


No. of Repet. 


Time Spent 


No. of Repet. 


1917 










Mon. Oct. 29 


20 min. 


4 






Tues. Oct. 30 






40 min. 


8 


Wed. Oct. 31 






30 min. 


6 


Thurs. Nov. 1 


20 min. 


4 






Fri. Nov. 2 






20 min. 


4 


Sat. Nov. 3 










Sun. Nov. 4 


20 min. 


4 






Mon. Nov. 5 






15 min. 


3 


Tues. Nov. 6 










Wed. Nov. 7 


20 min. 


4 






Thurs. Nov. 8 










Fri. Nov. 9 






10 min. 


2 


Sat. Nov. 10 


20 min. 


4 






Sun. Nov. 11 










Mon. Nov. 12 










Tues. Nov. 13 


20 min. 


4 


5 min. 


1 


Totals 


120 min. 


24 


120 min. 


24 



Thus we have, according to the plan of the experiment, an equal 
number of practice-periods spread over approximately (Reducing 
schedule one day shorter) the same total length of time. The total 
amount of time spent (120 min.), the average length of the study- 
periods (20 min.), and the total number of repetitions of the bonds 
(24) were the same for both schedules. 



RESULTS OF THE EXPERIMENT 



Two tests were given, the first on the day following the comple- 
tion of the practice, and the second one week later. None of the 
subjects except the author, however, knew that a second test was to 
be given. 



6 Improvement and the Distribution of Practice 

The order of the words in each list was rearranged for the purpose 
of the first test. List No. I, which was studied according to the 
Equal schedule, was given first, and after a few minutes intermission 
list No. 2, which was studied according to the Reducing schedule, 
was given. In the second test, the words were presented in the 
original order in which they were studied, but the order of giving 
the two lists was reversed, i. e., list No. 2 was given first, and after 
a few minutes' intermission list No. I was given. 

Three of the five subjects did better with the Reducing list in both 
tests; the other two consistently did better with the Equal list. 
The number of words attempted and the number recalled correctly 
by each of the subjects in both tests were as follows: 







FIRST 


TEST 




SECOND TEST 


Subject 


No. Attempted 


No. R 


^called 


No. Att 


empted 


No. Recalled 




Equal 


Reduc. 


Equal 


Reduc. 


Equal 


Reduc. 


Equal 


Reduc. 


1— Alb. 


133 


148 


127 


H5 


130 


135 


117 


125 


2 — Bar. 


H3 


H5 


140 


144 


136 


142 


129 


140 


3 — Cum. 


129 


145 


114 


130 


135 


142 


124 


133 


4 — Sen. 


95 


76 


78 


65 


95 


82 


75 


71 


5— Coo. 


in 


90 


103 


77 


117 


103 


93 


87 


Totals 


611 


604 


562 


56i 


613 


604 


538 


556 


Average 






112. 4 


112 .2 






107.6 


112. 2 


P. E. 






14.8 


27.9 






16.0 


21.8 



The small amount of forgetting during the interval of one week 
between the first and the second tests, as shown by the above figures, 
may be partly accounted for by reference to three facts: (1) three 
of the subjects were studying beginning French at the time of the 
experiment; (2) one of the subjects (the author) had the advantage 
of going over both lists of words several times during the interim 
between the tests, in scoring the results of the first test; (3) the 
rearranging of the order of the words in the first test caused a lower 
score than otherwise might have been made, and the presenting of 
the words in their original order in the second test tended to give 



Learning French Vocabularies by Adults 7 

a somewhat higher score than otherwise. Hence the amount of for- 
getting appears less than would have been the case had the condi- 
tions just described been reversed in the two tests, or even had they 
been made equal in both tests. 

The influence of these factors does not, of course, in any way 
invalidate the comparison of the two schedules of distribution of 
practice, as both lists of words were influenced alike. The only 
effect would be to reduce the amount of forgetting shown during the 
interim between the two tests. 

Combining the results of the two tests we have, for the five sub- 
jects, an average of no words recalled correctly for the Equal list 
and an average of 1 1 1 .7 words recalled correctly for the Reducing list. 
Thus there is an average difference of 1.7 words in favor of the Re- 
ducing list. Owing to the large variability in the differences be- 
tween the results of the two lists for the different subjects and the 
small number of subjects, the unreliability of this difference of 1.7 is 
very high. The P.E. of the t. — obt. Diff. is 3.56. This means that 
the chances are about even that the true difference with a large 
number of subjects will lie between 1.86, or about two per cent, in 
favor of the Equal schedule, and 5.26, or about six per cent, in favor 
of the Reducing schedule. 

DISCUSSION OF THE TWO SORTS OF DISTRIBUTION 

In general, the differences between these two sorts of distribution 
are as follows: (1) In respect to the length of the practice-period, 
the Reducing schedule has a larger number of repetitions occurring 
in the early part of the experiment, while with the Equal schedule 
the number of repetitions remains uniform for each practice-period 
throughout the experiment. (2) The length of the time 7 interval 
between periods gradually increases in the case of the Reducing 
schedule, while with the Equal schedule it remains constant. 

The actual differences between the two schedules in our experi- 
ment, together with certain suggestions as to the possible influence 
of same upon the results, appear as follows: (1) The total length 
of time covered by the experiment was substantially the same 
{Reducing schedule one day shorter) for both schedules. (2) The 
number of practice-periods (six) was the same in both cases. (3) 
The number of repetitions of the bonds (twenty-four) was the same. 
(4) The length of the practice-periods was uniform (twenty minutes) 



8 Improvement and the Distribution of Practice 

throughout for the Equal schedule, but for the Reducing schedule 
the length varied from forty minutes at the beginning of the experi- 
ment to five minutes at the close. (5) The time-interval between 
periods was uniform (three days) for the Equal schedule, while for 
the Reducing schedule it varied from one day at the beginning to 
four days at the close. In all other respects the conditions were 
substantially the same for both schedules. ( 

Referring to the length of the practice-period it might be argued 
by some that a forty-minute period is too long for grade pupils r 
especially for the lower grades, as fatigue may set in, or the pupils 
become bored by so long a period of intensive work. So, also, it 
might be argued that a five-minute period is too short to allow for 
a proper 'warming up'. If these suppositions be true, then our 
Reducing group suffered a disadvantage. 

On the other hand, the knowledge of a shorter practice-period at 
each succeeding time might, conceivably, serve as a stimulant. 
Moreover, the gradually decreasing length of the practice-period, to- 
gether with an increasing interval between may be said to approxi- 
mate more nearly the curve of forgetting; hence the Reducing 
schedule may, after all, be a more favorable arrangement. 

With regard to the time-interval between periods, it would seem 
fair to conclude that neither schedule had any very great advan- 
tage, as no interval was shorter than twenty-four hours and none 
was longer than four days. 

The optimum distribution of time has not, as yet, been determined 
in a sufficient number of cases to enable one to formulate any gen- 
eral rule that might be calculated to hold true for all situations. 
The whole matter is summed up by Thorndike in the following 
words : 

The experimental results obtained justify, in a rough way, the avoidance 
of very long practice-periods and of very short intervals. They seem to show, 
on the other hand, that much longer practice-periods than are customary in 
the common schools are probably entirely allowable, and that much shorter 
intervals are allowable than those customary between the first learning and 
successive 'reviews' in schools. 

What period-length shall be considered 'very long' depends on the amount 
of variety and satisfyingness the function shows. Two hours is, thus, a very 
long period for adding or learning 32-syllable nonsense series, but perhaps 
not for playing golf or chess. 



Learning French Vocabularies by Adults 9 

What interval between periods shall be considered 'very short' depends on 
the length of the periods themselves, and also on the character of the function. 
For adding practised in twenty-minute periods, an interval of five minutes 
would be very short, and probably also one of five hours. The knowledge 
that would enable one to define the statement made in the text is lacking. 

In a more recent work Hollingworth and Poffenberger confirm the 
above summary with the following statement : 

The conclusion which has been drawn from experiments of this type is 
that too great concentration or distribution of time is not economical. The 
learning periods should be short enough to avoid the onset of fatigue, and 
long enough not to cause the loss of too much time in getting warmed up to 
the task at the beginning of the learning period. 

According to Jost's law, stated in the Introduction, one would 
expect our Reducing schedule to show better results than the Equal, 
because of the piling up of repetitions in the earlier part of the 
experiment, which in turn would mean a larger number of bonds 
that were older when the same were exercised again later on in the 
experiment. Just how to equate this fact against the further fact 
that these older bonds would necessarily receive a fewer number of 
repetitions as compared with bonds of equal age in the Equal 
schedule, however, is a problem yet unsolved. It should, at least, 
be remembered that Jost's main argument was based upon the re- 
learning of associations which had been formed up to a given 
strength — that of producing one correct recall — and subsequently 
dropping below the limen through disuse, while most of the experi- 
mental work bearing upon the question of length of practice-period 
and time-interval between has been done by the exercising of sub- 
liminal bonds. 

The results of our experiment showed a small difference in favor 
of the Reducing schedule. The large variability amongst individ- 
uals in the differences between the two schedules, as shown by the 
results obtained, together with the small number of subjects, gives 
the very high unreliability index of 3.56. This means that the 
chances are about four to one that the true difference with a large 
number of subjects will lie between 5.42 in favor of the Equal 
schedule and 8.82 in favor of the Reducing. The chances are about 
twenty-four to one that the true difference will lie between 8.98 
in favor of the Equal schedule, and 12.38 in favor of the Reducing, or 
about ninety-nine to one that the true difference will lie between 



I o Improvement and the Distribution of Practice 

12.54 i n favor of the Equal schedule and 15.94 in favor of the Re- 
ducing schedule. 

In the succeeding chapters will be found a report of an attempt to 
further answer this same question through a series of experiments 
with public school children working under ordinary everyday con- 
ditions. The same problem was also studied with a group of 157 
normal school students acting as subjects. On account of certain 
difficulties arising from the attempt to use entirely new material in 
the experiment with school children, it was necessary to adapt 
material from that which is regularly used in the schools. This was 
done by making use of material taken from geography, history, ad- 
dition, and division. A part of this same material was also used in 
the experiment with normal school students. 



SECTION II 
THE LYNDHURST EXPERIMENTS 

I. DESCRIPTION 

In order to avoid certain repetitions the experiments in learning 
geographical and historical facts and in practising addition and 
division will be described, not by topics, but by groups of individuals 
concerned. 

CHARACTER OF THE SCHOOL POPULATION 

Nationality. The Lyndhurst public schools at the time of our ex- 
periment (1914-15) had an enrollment of about 1,500 pupils in the 
grades, of which number approximately fifty per cent were foreign- 
born, being chiefly Italians and Poles. Taken altogether, we should 
say the population is somewhat below that of an average city of this 
size, in respect to the use of English. 

Economic conditions. Because of the nature of the occupations 
followed by the citizens (railroad shops, New York business posi- 
tions, etc.) there are few families that are either very poor or very 
well-to-do. All in all, the community would probably rank below, 
say, an average rural town of the same size in respect to general 
economic conditions. 

Educational conditions. From the standpoint of equipment, the 
school system is fairly well provided for. The work done in the grades 
is accredited by the neighboring high schools. The teaching staff, 
composed of forty-two women, was well recommended and in charge 
of a competent supervising principal. As to the relative standing of 
the pupils in such functions as were tested, the reader is referred to 
the summary of all our experiments found at the close of Section IV. 

In the Rutherford schools, from which we drew the seventh grade 
subjects for our experiment, there are few foreigners and more well- 
to-do homes. Here also we found a competent superintendent with 
an excellent corps of teachers. 

SUBJECTS USED IN THE EXPERIMENT 

The subjects consisted of 114 pupils from the seventh grade, 130 
from the sixth grade, 127 from the fifth grade, 178 from the fourth 



12 Improvement and the Distribution of Practice 

grade, and 150 from the third grade, making a total of 699 whose 
records were complete enough to count. The records of those who 
happened to be absent on either of the test days, together with those 
who were transferred in or out while the experiment was going on, 
were eliminated because of incompleteness. 

CONTENT-MATERIAL USED 

Geography. The material used for practice in geography was de- 
vised by the author. It consisted of condensed facts and principles 
printed on sheets six by ten inches in dimension. There were two 
sheets of a kind, one being used for study and the other for the test, 
the only difference being that on the test sheets certain important 
words were arbitrarily omitted. 

The following are samples of the material : 

Principles of Geography 
(study sheet) 

1. Geography is the (1) relation of the earth's surface to the (2) plants and 

(3) animals living upon it. 

2. The earth's (4) crust is composed mostly of (5) rocks and detritus. 

3. The earth is surrounded by a (6) mixture of oxygen, (7) nitrogen, and 

other gases called (8) air. 

4. The larger portion of the earth's (9) surface is covered with (10) water, 

which is composed of two parts of (11) hydrogen to one part of (12) 
oxygen. 

Principles of Geography 

(test sheet) 

1. Geography is the (1) of the earth's surface to the (2) and 

(3) living upon it. 

2. The earth's (4) is composed mostly of (5) and detritus. 

3. The earth is surrounded by a (6) of oxygen, (7) and 

other gases called (8) 

4. The larger portion of the earth's (9) is covered with (10) , 

which is composed of two parts of (11) to one part of (12) 

Geography of South America 

(study sheet) 

South America as a continent ranks (1) fourth in size, is (2) less important 
in commerce than Europe, is shaped most like (3) North America, is 
(4) moist and hot in climate, except on the (5) western coast. 



The Lyndhurst Experiments with Grade Pupils 13 

South America has (6) few good harbors, the best ones being in the country 
of (7) Chile, the chief mountains are the (8) Andes, and the great eastern 
expanse is called the (9) Brazilian highlands. Between this region and the 
(10) Guiana highlands is an extensive lowland drained by several large rivers, 



Geography of South America 

(test sheet) 

South America as a continent ranks (1) in size, is (2) important 

in commerce than Europe, is shaped most like (3) , is 

(4) and hot in climate, except on the (5) coast. 

South America has (6) good harbors, the best ones being in the country 
of (7) , the chief mountains are the (8) , and the great eastern 

expanse is called the (9) highlands. Between this region and the 

(10) highlands is an extensive lowland drained by several large rivers, 

Geography of Europe 
(study sheet) 

Europe is the smallest continent in the world, except (1) Australia, has a 
very (2) irregular coast line, with (3) many good harbors, produces crops 
very like (4) North America, has a well-developed system of (5) commerce 
both by land and by (6) water, the people belong mostly to the (7) white 
race, who engage chiefly in (8) agriculture in the plains, (9) fishing along 
the coast, (10) hunting in the tundra area, with extensive manufacturing in 

Geography of Europe 
(test sheet) 

Europe is the smallest continent in the world, except (1) , has a 

very (2) coast line, with (3) good harbors, produces crops 

very like (4) , has a well-developed system of (5) 

both by land and by (6) , the people belong mostly to the (7) 

race, who engage chiefly in (8) in the plains, (9) along 

the coast, (10) in the tundra area, with extensive manufacturing in 

History. For practice in history, the material was arranged after 
the same manner as that described for geography. The subject- 
matter, however, was taken from the summaries found at the end 
of the chapters in Montgomery's Beginners History of America. 
The author is responsible for the arrangement and adaptation of 



14 Improvement and the Distribution of Practice 

the text to suit the purpose of the experiment. There were five 
sheets of this material printed, the following being sample pieces of 
sheet No. I : 

American History I 

(study sheet) 

About the year (i) iooo, (2) Leif Ericson, a Northman, discovered Amer- 
ica. He called it (3) Vinland. The (4) Northmen did not stay in America, 
and the discovery did them no real good. 

Nearly (5) 500 years later, Christopher Columbus, an (6) Italian, set out 
from (7) Spain to find a short (8) direct way to the (9) Indies by water. 
He crossed the (10) Atlantic and discovered (11) America in (12) 1492. 
But he thought the land he had seen was part of (13) Asia. He called it the 

American History I 
(test sheet) 

About the year (1) , (2) , a Northman, discovered Amer- 

ica. He called it (3) . The (4) did not stay in America, 

and the discovery did them no real good. 

Nearly (5) years later, Christopher Columbus, an (6) , set out 

from (7) to find a short (8) way to the (9) by water. 

He crossed the (10) and discovered (11) in (12) 

But he thought the land he had seen was part of (13) . He called it the 

Division. For practice in this subject, the short division examples 
devised by Thorndike and rearranged by Kirby were used. In order 
to prevent the memorizing of the answers, three different sheets 
containing equal tasks were used. Each sheet contained approxi- 
mately ninety-six examples, of which the following are samples : 

6s and r. 

6s and r. 

4s and r. 

8s and r. 

9s and r. 

7s and r. 

7s and r. 

9s and r. 

Addition. For practice in addition, the standard single-column 
addition sheets of Thorndike were used. By the use of four different 
sheets, each containing tasks of equal difficulty, the possibility of 



20 = 


5s 




3i = 


7s and 


r. 


22 = 


56= 


9s and 


r. 


83 = 


9s and 


r. 


53 = 


30 = 


7s and 


r. 


21 = 


7s 




33 = 


89= 


9s and 


r. 


54 = 


8s and 


r. 


77 = 


20 = 


8s and 


r. 


32 = 


4s 




22 = 


56= 


6s and 


r. 


80 = 


9s and 


r. 


52 = 


31 = 


4s and 


r. 


22 = 


3s and 


r. 


33 = 


86 = 


9s and 


r. 


53 = 


9s and 


r. 


75 = 



The Lyndhurst Experiments with Grade Pupils 15 

memorizing the answers was reduced to a minimum. The following 
is a sample third of a sheet: 

233949 3 . 298767569 
5572237547925924 
8737368644692577 



s 


2 


8 


8 


2 


6 


5 


8 


7 


9 


4 


6 


7 


6 


5 


8 


7 


6 


2 


2 


9 


5 


3 


3 


6 


4 


9 


5 


5 


8 


2 


8 


7 


8 


6 


4 


5 


8 


2 


8 


5 


8 


7 


6 


8 


2 


9 


2 


2 


7 


9 


5 


8 


4 


8 


8 


9 


4 


5 


3 


6 


9 


8 


9 


9 


9 


3 


7 


7 


8 


3 


7 


4 


7 


9 


5 


4 


7 


4 


2 


6 


4 


6 


4 


5 


3 


9 


3 


4 


9 


8 


9 


8 


7 


2 


7 


9 


9 


8 


9 


5 


2 


8 


8 


8 


7 


6 


7 


3 


3 


7 


6 



PLAN OF THE EXPERIMENT 

Our general plan was to conduct the experiment under ordinary 
school conditions, equalizing all the factors save that of distribution 
of practice. The general principle followed in the arrangement of 
the two schedules was that discussed in the previous section. This 
necessarily involved differences in the length of the practice-periods 
and in the intervals between the periods. In most cases there was 
also a difference in the number of practice-periods. 

Table II shows the exact arrangement of the time-schedule for 
the four sixth grade classes which learned geography and history. 
Table III exhibits similar data for the five seventh grade classes 
which learned principles of geography. Table IV gives the data for 
the eleven third and fourth grade classes which practised addition 
and the four fifth grade classes which practised short division. 



16 



Improvement and the Distribution of Practice 



TABLE II 

THE NUMBER OF MINUTES DEVOTED TO 
EACH PRACTICE-PERIOD IN THE CASE 
OF CLASSES 'P'.AND 'q' WHICH LEARNED 
GEOGRAPHICAL FACTS, AND CLASSES V 
AND 's', WHICH LEARNED HISTORICAL 
FACTS 

These were sixth grade classes from 
the Lyndhurst schools. 



TABLE III 

THE NUMBER OF MINUTES DEVOTED TO 
EACH PRACTICE-PERIOD IN THE CASE 
OF CLASSES 'T\ 'u\ V, 'w', AND 
'x', WHICH LEARNED GEOGRAPHICAL 

FACTS 

These were seventh grade classes from 
the Rutherford schools. 



1915 


P 


q 


r 


s 




1915 


t 


u 


V 


w 


X 


Feb. 


6 


6 


6 


6 




Mch. 


7 


7 


7 


7 


7 


9 


20 


20 


20 


20 


5 


15 


15 


15 


15 


15 


10 












6 












11 




15 




15 




7 












12 


20 




20 






8 


15 


15 


15 


15 


15 


13 












9. 












14 












10 


15 


15 


10 


10 


10 


15 




12^ 




I2# 




11 












16 


20 


1254 


20 


I2>^ 




12 


15 


15 


10 


10 


10 


17 




10 




10 




13 












18 


20 


10 


20 


IO 




14 












19 




lA 




IV* 




15 












20 












16 












21 












17 


15 


15 


15 


15 


15 


22 


20 


VA 


20 


lA 




18 












23 




5 




5 




19 






10 


10 


10 


24 


20 


20 


20 


20 




20 
21 
























22 


15 


15 


10 


10 


10 














23 
























24 






7K 


7K 


1A 














25 
























26 


15 


15 


VA 


VA 


lA 














27 
























28 
























29 






5 


5 


5 














30 
























3i 


15 


15 


15 


15 


15 



The Lyndhurst Experiments with Grade Pupils 
TABLE IV 



17 



THE NUMBER OF MINUTES DEVOTED TO EACH PRACTICE-PERIOD, IN THE CASE OF 

CLASSES 'A' TO 'k', INCLUSIVE, WHICH PRACTISED SINGLE-COLUMN 

ADDITION AND CLASSES 'l', 'm', 'n,' AND 'o', WHICH 

PRACTISED SHORT DIVISION 

These were third, fourth, and fifth grade classes from the Lyndhurst schools. 



1914 
Nov. 



15 



i5 
5 

5 



15 

[5 



IS 
15 



15 



15 



15 

5 
5 
5 



15 

5 
5 
5 



15 



15 



15 
15 



15 



15 



15 



15 



15 



•5 



15 



1,5 



9 
10 
11 
12 
13 

15 
16 

17 
18 

19 
20 
21 
22 

23 
24 

25 
26 

27 
28 
29 
30 
D.i 

2 

3 

4 



15 

7A 



7A 



7A 



5 
15 

7A 



2^ 



15 

7K 



^A 



15 

7^ 



15 



10 
15 



5 

5 

5 

10 



5 
15 



5 

5 

5 

10 



5 
15 



15 
15 



15 
15 



15 
15 



10 
10 



rA 



VA 



7A 



15 



*A 

15 



2^ 

15 

lA 



2J 
15 

7^ 



15 



10 
10 



i8 



Improvement and the Distribution of Practice 
TABLE IV {Continued) 



1914 
Dec. 



5 
6 

7 
8 

9 
10 

11 
12 
13 
14 
15 
16 

17 

18 



■14 



15 



15 



2^ 
15 



2K 
15 



15 



15 



15 



15 



15 
15 



15 



15 



15 



2K 
15 



m 
5 



15 



15 



ADMINISTRATION OF THE EXPERIMENT 

The author was accorded the most favorable advantages for carry- 
ing out the experiment. Full permission was granted to interrupt 
classes at any hour, the class being turned over by the teacher imme- 
diately upon our entering the room. Whenever possible, the practice 
was done at the regular recitation period for the subject that was to 
be used. Thus, in the experiment with geography and history, the 
practice was done at the regular recitation period for these subjects. 

Method of conducting the experiment. All the initial and final tests, 
as well as all the intervening practice-periods, with but few exceptions, 
were conducted by the author. In all cases where assistance was ren- 
dered, the teachers were thoroughly familiarized with the work be- 
forehand ; hence, there is probably no appreciable difference in the re- 
sults due to the small amount of teacher-assistance that was used. 

The method of procedure used in all the rooms was as follows: 
As soon as we had entered the room the teacher had all the work 
cleared from the pupils' desks and immediately turned the class over 
to us, she herself either leaving the room or, if remaining, assuming 
the attitude of one of the group. 

• The first step was to explain and illustrate the nature of the work 
to be done. It was found that this could best be done by placing 



The Lyndhurst Experiments with Grade Pupils 19 

before the pupils a sample of the work and allowing them to try it 
for themselves. In the case of those classes which learned geography 
and history the study sheets were first passed out, the only instruc- 
tions necessary being simply to say, "We are going to study this 
text a little while, and the way we are going to study it is simply to 
read it over in concert, all following the leader. I will be the leader. 
All ready, Read!" 

After having read over a few of the sentences, attention was called 
to the words immediately following the numbers, as being the most 
important ones to remember. Having put in the required amount 
of time in study, these sheets were collected and the test sheets, 
together with blank sheets on which to write the words, were passed 
out. The pupils were instructed to read over the sentences on the 
test sheets, endeavoring to recall as many of the omitted words as 
possible, and to write them in order on the blank sheets of paper 
provided. They were admonished not to spend too much time in 
trying to think of any particular word. In case a word could not be 
recalled within, say, five or ten seconds, they were simply to draw a 
line instead on the scoring sheet and pass on to the next, the object 
being to get down as many words as possible within the time. 

By preliminary testing with adults, it was found that approximately 
seven minutes were required to read over two of the study sheets. It 
was also judged that two sheets, which contained on an average about 
seventy test words each, would be sufficient to measure the ability 
of the best pupils, in a test not to exceed twenty minutes in length. 
Two sheets, therefore, were taken as the unit of material to be 
learned. By this arrangement, the bonds to be tested were exercised 
on an average from fourteen to seventeen times during the experiment. 

In initiating the practice with addition and division, a sample piece 
of a sheet was placed before each pupil, and, after a brief illustration 
on the blackboard, the following general directions were given: 
"When the signal is given you are to begin and add (or divide, as the 
case might be) as many of the examples as you can until you are told 
to stop. You are to have about three minutes. The signal to begin 
will be 'Ready, Go!' The signal to stop will be 'Pencils down!' You 
must work as hard as you can, but don't make any mistakes." 

The trial was made, the trial papers were taken up, new sheets 
were distributed with the figures down, whereupon the experi- 
menter asked, "When the signal is given, what are you to do?" 



20 Improvement and the Distribution of Practice 

To which the class replied in concert, "Turn over the papers, work 
as hard as we can, and don't make any mistakes." "Very well," 
said the experimenter, "you are to have five minutes this time; all 
ready, Go!" and the real experiment was begun. 

In the case of single-column addition, such as was used, there are 
thirty-seven possible bonds that may be exercised. Since there are 
ten digits to a column, each bond occurs by chance once in every 
four columns. Thus the bonds tested were exercised on an average 
from ten to fifty or more times each during the course of the experi- 
ment, depending upon the total number of columns added. 

Incentives used. The incentives used in general were those ordi- 
narily used in school work, viz., class pride, individual pride, small 
material rewards, and the idea of competing with one's own past 
record. This last-named incentive, however, was made more em- 
phatic than it is ordinarily made by the teacher. 

The appeal to class pride was made by announcing to each class 
that a comparison was to be made of the records of the different 
classes. Personal pride was called forth by reporting from day to 
day the names of those pupils who showed improvement. The 
material rewards given consisted of a silver half-dollar to the pupil 
who showed the largest per cent of individual improvement, and 
a silver quarter-dollar to the one who showed the next largest 
per cent of individual improvement, in both the third and fourth 
grades. The idea of competing with one's own record was illus- 
trated from time to time, and the indications were that a majority 
of all the pupils understood it and were stimulated by it. Indeed, 
many were very much influenced by this form of incentive, as was 
evidenced by the fact that they kept track of their own improve- 
ment from day to day. It is not claimed that our pupils were stim- 
ulated to any unusual degree, for the intention was to get only such 
results as might reasonably be expected of any similar set of pupils 
working under like conditions. Our pupils worked, we should say, 
as faithfully as the average pupils work in any ordinary school sys- 
tem under ordinary everyday conditions. 

SCORING AND TABULATING THE RESULTS 

Geography and history. The score was the number of words re- 
membered and correctly written. The per cent of words attempted 
that was found correct was taken as the index of accuracy. Inasmuch 



The Lyndhurst Experiments with Grade Pupils 2 1 

as most of the time was taken up with the study of the text, no 
attempt was made to keep a daily record of the scores. 

Since a pupil would sometimes apparently understand the mean- 
ing of a sentence, but could not recall the exact word that had been 
omitted from the test sheet, it was necessary to standardize the 
various substitutions that were made. This was accomplished by 
assigning each substituted word to one of three classes, designated 
as Full credit (F. C), Half credit (H. C), or Rejected (R.). These 
lists constituted a guide for the uniform scoring of all the papers in 
geography and history. 

Below are given some of the more interesting substitutions made by 
various pupils, classified according to the probable explanation of each. 

1. Substitutions apparently due to the lack of knowing how to spell the word 
correctly: "lamor," "larnma" for "llama," F. C. ; "brazla" for "Brazil," H. C. ; 
"guaquil" for "Guayaquil," F. C; "Dublon" for "Dublin," F. C; "Carcas" for 
"Caracas," F. C; "Venemeeda" for "Venezuela," R. 

2. Substitutions evidently due to a confusion of words: "Tampico" for "tapi- 
oca," R.; "area" for "rhea," R.; "revelations" for "revolutions," R.; "chan- 
nels" for "canals," H. C. 

3. Substitutions of words having similar meanings: "some" for "few," F. C; 
"lowlands" for "plains," F. C. ; "wrote" for "published," H. C. ; "ways" for 
"habits," H. C; "liberty" for "freedom," F. C. 

4. Substitutions of words with similar meanings but in which case the dis- 
tinction is rather more important than in (3) above: "everlasting" for "con- 
stant," F. C. ; "red" for "copper," F. C, when referring to the color of the 
Indians, but "brown" for "copper," R., because this would not distinguish the 
Indians from certain other races. 

5. Substitutions representing different degrees of comprehensiveness of 
thought: "U. S." for "America," F. C; for pupils in the fifth and sixth grades, 
but only H. C. for seventh and eighth grades. "Palos" for "Spain," F. C, 
when designating the place from which Columbus sailed; "battles" for "wars," 
F.C.; "fur" for "hair," H.C.; "meat" for "mutton," H. C; "steel" for "mag- 
net," R. ; "housework" for "work," R., the reference being to the Indian 
squaw, who does most of the "work." 

Addition and division. In the case of these two subjects all the 
time allotted to each practice-period was used in solving as many 
of the examples as possible, hence it was possible to keep complete 
daily scores. The score was the number of examples solved cor- 
rectly. The accuracy was recorded as: 

number of problems done correctly 
number of problems attempted 



22 



Improvement and the Distribution of Practice 



Samples of the daily records of the two classes are given below. 
The numbers down the left-hand margin are for the identification 
of the pupils. The letters 'A' and 'E' at the top refer, respectively, 
to the number of examples attempted and to the number of errors 
made. The figures above the letters represent the number of min- 
utes devoted to each practice-period. 



SAMPLE NO. I, FROM THE DAILY RECORD OF CLASS "f," 
WHICH FOLLOWED THE Equal SCHEDULE 



Boj's 


15 


10 


IO 


10 


10 


10 


10 


10 


15 


15 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


i 

2 

3 
4 
5 
6 

7 
8 

9 

IO 


18 
17 

20 

13 

28 

3i 
15 
16 

20 


4 

8 
3 

2 

14 
16 

13 
3 
4 


12 

8 

15 

18 

20 

27 

39 
16 

20 

27 


5 

2 
10 

4 

5 

21 

39 
6 

8 

5 


18 
8 

14 
4i 
19 

32 

24 
10 
16 
23 


13 
3 
8 

36 
3 

32 

22 

9 
8 
6 


18 

8 

12 

31 

18 

19 
32 
23 


1 

1 

6 

11 



7 

9 

12 


27 

8 

16 

19 
11 

30 
26 

14 
23 


3 
1 

4 





20 

23 
1 
6 


9 
16 
11 

15 
27 
16 
12 

25 


1 
8 

I 
11 
12 
4 

7 


13 
12 
11 
22 
15 
37 

13 
H 
3i 


1 

4 

8 

18 

3 
21 

3 

6 

12 


23 
12 

13 

47 
15 
28 

25 

9 

16 

24 


2 

3 
10 

46 

4 
20 

7 
2 

10 

7 


28 

16 

48 
18 
44 
43 
24 
21 

39 


8 

12 

45 
4 
32 
34 
23 
11 
12 


32 

3i 
48 
30 
44 
48 

48 


7 

14 

25 

4 

30 

30 

18 



SAMPLE NO. 2, FROM THE DAILY RECORD OF CLASS "c," 

which followed the Reducing SCHEDULE 



Girls 


15 


15 


10 


10 


7^ 


5 


2K 


1 


5 


7K 


5 


5 


2K 


15 




A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


A 


E 


1 


31 


27 


19 


12 


13 


12 


13 


6 


5 


3 


3 





2 


2 


14 


3 


11 


5 


7 


3 


11 


8 


3 


I 


20 


16 


2 


35 


6 


38 


3 


26 


14 


18 


2 


14 


1 


9 





7 


1 


21 


1 


17 





16 


2 


15 





5 





40 


8 


3 


9 


7 


13 


7 


10 


6 


9 


2 


8 


4 


7 


2 


3 


1 


17 


6 


8 


6 


7 


5 


8 


6 


4 


3 


19 


12 


4 


27 


5 


28 


5 


21 


6 


13 


3 


16 


3 


14 


3 


8 


1 


29 


8 


19 


2 


12 


3 


12 


2 


6 









5 


14 


4 


19 


5 






10 


2 


10 


2 


9 


1 


4 


2 


20 


5 


13 


4 


7 


1 


9 


1 


4 





24 


5 


7 


27 


7 


24 


8 


18 


15 


20 


6 


15 


3 


13 


2 


6 


1 


33 


5 


15 


7 


9 


4 


11 


2 


6 





30 


15 


8 


18 


6 


31 


15 


15 


7 


12 


4 


13 


3 


16 


6 


3 


1 


16 


7 


14 


3 


9 


1 


9 


2 


4 


1 






11 


15 


6 


19 


6 


17 


6 






4 





4 


1 


2 


1 


8 


1 


13 


4 


















12 


12 


9 


15 


8 


6 


4 


6 


3 


7 


3 


6 


2 


5 


1 


15 


5 


10 


3 


7 


3 


8 


3 


5 


1 


16 


12 


13 


9 


4 


15 


5 


13 


5 


15 


2 


11 





7 


1 


3 





19 


1 


IC 


1 


6 


2 


6 


4 


3 





20' 


7 



The Lyndhurst Experiments with Grade Pupils 23 

In scoring the daily practice done in addition and division, the 
assistance of three grammar grade boys was employed. They were 
paid nominal wages for the work, which was done on Saturdays and 
other school holidays. The work was done under the supervision of 
the author, who himself did the scoring of the initial and the final 
tests, also all the tabulating of the results. The work of scoring the 
papers was reduced to a mere matter of comparison, by the use of 
'key sheets'. The 'key sheets' contained the correct answers so 
arranged that, when superimposed upon the test sheets, the correct 
answers appeared alongside of the answers put down by the pupils. 
Thus it was an easy matter to check the wrong answers. The ac- 
curacy of the work done by the helpers was checked from time to 
time by the author. 

All the original daily records for addition and division in both the 
Lyndhurst and the Ohio experiments were preserved. These orig- 
inal records furnish a mass of data that is valuable for the study of 
various other problems, such as individual differences, individual 
learning curves, sex differences, etc. In view of the fact that most 
of the time spent in the geography and history experiments was 
taken up in study, the daily scores for these subjects were of little 
significance, and were not preserved. 

2. RESULTS OF THE LYNDHURST EXPERIMENTS 

This section will exhibit the results of the Lyndhurst experiments 
in the form of tables of frequencies. The frequencies given are in 
terms of the initial scores, the per cent of problems correct in the 
initial test, and the change that took place in these two respects, as 
a result of the 100 minutes of practice that was measured. 

TABLES OF FREQUENCIES 

Table V gives the frequencies for the initial scores made by the 
Equal and the Reducing groups. The comparison is made grade by 
grade, except in the case of the third and fourth grades, the data 
for which grades are combined. Table VI gives the frequencies for 
the per cent of problems solved correctly in the initial test. Table 
VII gives the frequencies for the amount of change that took place, 
in the number of problems correctly solved, during the time of the 
experiment. Table VIII gives the frequencies for the change that 
took place in respect to the per cent of problems solved correctly. 



24 Improvement and the Distribution of Practice 

The total amount of time consumed in each of the geography and 
history experiments was 120 minutes. The initial score was counted 
as the average, or mid-point, of the first twenty minutes of practice, 
and the final score was counted as the average, or mid-point, of the 
last twenty minutes of practice. Hence, the amount of practice that 
was measured was that which lies between these two mid-points, or 
100 minutes. 

The total amount of time consumed in each of the addition and 
division experiments was 115 minutes. In this case, however, the 
initial and final scores were counted at the average, or mid-point, of 
the first and last fifteen minutes of practice, respectively, hence the 
amount of practice that was measured was again just 100 minutes. 

The results of the geography and history experiments were influ- 
enced by two facts. First, a few minutes of study were allowed 
before the initial test was given, in order that the pupils might 
become acquainted with the nature of the material. Thus there is 
included in the initial score a slight effect due to immediate memory. 
This fact, as may readily be seen, has the tendency to make the 
initial score higher than it otherwise would have been, and to make 
the amount of gain shown during the time of the experiment less 
than it otherwise would have been. Second, there was no study 
immediately preceding the final test, hence the amount of gain re- 
ported is still further decreased below what it would have been, 
with an equality of these conditions. 1 

1 This does not, of course, affect the comparison of the two sorts of distribution of 
practice, but only the amount of gain shown during the experiment as a result of the 
100 minutes of practice with the functions concerned. 



The Lyndhurst Experiments with Grade Pupils 



25 



TABLE V 

INITIAL ABILITY IN THE LYNDHURST EXPERIMENTS: 
INITIAL SCORES 



FREQUENCIES FOR 



The figures at the top represent the number of problems solved correctly in the initial test, which 
was 20 minutes in the case of geography and history, and is minutes in the case of addition and 
division. 

The table reads as follows: Of the 56 seventh grade pupils who learned geographical facts according 
to the Equal schedule, there were none who made scores of o, plus 1, 2, 3 .... 9; 2 made scores of 10, 
11, 12, 13, or 14; none made scores of 15, 16, 17, 18, or 19; 10 made scores of 20, 21, 22, or 23, 24, etc. 



Time 
Dist. 


No. of 
Pupils 


Grade 


Material 




4 


5 
9 


10 

14 


15 
19 


20 
24 


25 
29 


30 
34 


35 
39 


40 

44 


Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 


56 

58 

30 
31 

34 
35 

56 
7i 

175 
153 


7 
7 

6 
6 

6 
6 

5 
5 

3,4 
3.4 


Geog. 
Geog. 

Hist. 
Hist. 

Geog. 
Geog. 

Div. 

Div. 

Add. 
Add. 


3 

25 
34 


1 

34 
39 


2 
1 

2 
1 

53 
34 


1 

2 
2 

1 

3 

32 
21 


10 
10 

5 
1 

2 

3 
1 

20 
15 


13 
11 

9 
3 

1 

8 

2 

4 

6 

4 


20 
13 

6 

4 

3 
3 

5 
3 

3 

5 


7 
11 

2 
6 

4 

7 

3 
1 

1 
1 


3 
6 

9 
6 

9 

4 

2 

4 

1 



{Above Table continued) 



Time 
Dist. 


45 
49 


50 
54 


55 
59 


60 
64 


65 
69 


70 
74 


75 
79 


80 
84 


85 
89 


90 
94 


95 
99 


100 
109 


no 
119 


120 
129 


130 

139 


140 
149 


150 
159 


Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 


1 

3 

1 

2 

5 
3 

3 


2 

1 

4 

6 

5 

3 

2 


1 
1 

4 
1 

6 

5 


2 

2 
1 

1 

7 


4 
4 


5 
2 


2 
3 


4 
5 


3 


1 

4 


1 
1 


2 

5 


3 
2 


3 
3 


2 


3 


1 
1 



26 



Improvement and the Distribution of Practice 



TABLE VI 



INITIAL ABILITY IN THE LYNDHURST EXPERIMENTS: FREQUENCIES FOR 
PER CENT OF ACCURACY 
The figures at the top represent the per cent of problems solved correctly of all that were attempted 
in the initial test. 

The table reads as follows: Of the 56 seventh grade pupils who learned geographical facts, accord- 
ing to the Equal schedule, there were none who showed as low a per cent of accuracy in the initial 

test as o, 1, 2, 3 49; one fell as low as so, Si. 52, S3, or 54 per cent; 1 fell as low as 

55. 56, 57, 58, or 59; 3 showed 60, 61, 62, 63, or 64 per cent, etc. 



Time 
Dist. 


No. of 
Pupils 


Grade 


Material 




4 


5 
9 


10 
14 

1 

3 
3 


15 
19 


20 

24 


25 
29 


30 35 
34 39 


40 

44 


Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 


56 

58 

30 
31 

34 
35 

56 
7i 

175 
153 


7 

7 

6 
6 

6 
6 

5 
5 

3,4 
3.4 


Geog. 
Geog. 

Mist. 
Hist. 

Geog. 
Geog. 

Div. 
Div. 

Add. 
Add. 


2 

9 
9 


1 

3 

4 










1 

1 


1 

2 


4 
4 


4 
11 


1 

7 
2 


7 
9 


1 

4 
14 











{Above 


Table t 


;on tinned) 










Time 


45 


50 


55 


60 


65 


70 


75 


80 


85 


90 


95 


100 


Dist. 


49 


54 


59 


64 


69 


74 


79 


84 


89 


94 


99 




Equal 




1 


1 


3 


4 


10 


11 


11 


11 


4 






Reduc. 




2 


6 


1 


11 


3 


12 


14 


6 


3 






Equal 


1 




2 


4 


5 


3 


2 


6 


4 


1 


1 




Reduc. 








3 


3 


4 


5 


7 


3 


5 






Equal 




1 


1 


2 


4 


2 


10 


8 


6 








Reduc. 




2 


1 


4 


4 


5 


9 


9 


1 








Equal 






1 




1 




4 


6 


12 


15 


13 


3 


Reduc. 


1 




1 




1 


4 


5 


4 


6 


H 


25 


5 


Equal 


6 


n 


13 


22 


14 


9 


11 


J 3 


18 


7 


6 


3 


Reduc. 


2 


14 


8 


11 


10 


14 


10 


9 


10 


4 


1 


2 



The Lyndhursl Experiments with Grade Pupils 



27 



TABLE VII 



AMOUNT OF IMPROVEMENT IN THE LYNDHURST EXPERIMENTS: 
TOTAL SCORE 

The figures at the top represent the amount of improvement, plus or minus, that took place during 
the time of the experiment in the number of problems correctly solved. 

The table reads as follows: Of the 56 seventh grade pupils who learned geographical facts accord- 
ing to the Equal schedule, there were none who made an improvement of minus 25, 24, 23 1 ; 

none showed an improvement of o, plus 1, 2, 3 9; 1 showed an improvement of plus 10, 

11, 12, 13, or 14; s showed plus 15, 16, 17, 18, or 19, etc. 



Time 
Dist. 


No. of 
Pupils 


Grade 


Ma- 
terial 


-25 
21 


-20 
16 


-15 
11 


-10 
6 


-5 
1 




4 


5 
9 


10 
14 


15 
19 


20 
24 


25 
29 


30 
34 


35 
39 


Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 


56 
58 

30 
31 

34 

35 

56 

7* 

175 
153 


7 
7 

6 
6 

6 
6 

5 
5 

3,4 
3,4 


Geog. 
Geog. 

Hist. 
Hist. 

Geog. 
Geog. 

Div. 
Div. 

Add. 
Add. 


1 


1 

1 


1 
1 


3 

3 

1 
1 

9 

7 


3 
1 

34 
17 


1 

5 

4 
2 

2 

44 
47 


1 

6 
I 

2 

2 

1 

33 
30 


1 

3 
1 

4 
3 

2 

24 

25 


5 

4 
1 

5 
3 

3 

15 
15 


6 
1 

1 

7 
6 

3 
2 

11 
6 


7 
5 

3 

1 
3 

2 
3 


8 
2 

6 

4 
4 

2 
I 

I 


2 

8 

2 

4 

4 
5 

1 

4 

1 



{Above Table continued) 



Time 
Dist. 


40 

44 


45 
49 


50 
54 


55 
59 


60 
64 


65 
69 


70 
74 


75 
79 


80 
84 


85 
89 


90 
94 


100 
109 


no 
119 


120 
129 


130 

139 


140 
149 


150 
159 


Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 


5 
6 

6 

I 

1 

7 

1 


8 

7 

2 
4 

1 

3 

3 


2 

5 

2 

2 

2 

3 


3 
6 

3 

3 

7 
4 


2 
6 

I 

2 
6 


I 
2 

1 

1 

4 


1 

2 

6 
2 


2 
2 

5 
6 


1 

2 

1 

3 


2 
2 

2 
1 


2 
6 


3 

4 


1 
1 


1 

3 


2 
3 


1 

2 


1 



28 



Improvement and the Distribution of Practice 



TABLE VIII 

AMOUNT OF IMPROVEMENT IN THE LYNDHURST EXPERIMENTS: 
PER CENT CORRECT 

The figures at the top represent the amount of improvement, plus or minus, in the per cent of prob- 
lems correctly solved, of all that were attempted, that took place during the time of the experiment, 

The table reads as follows: Of the 56 seventh grade pupils who learned geographical facts accord- 
ing to the Equal schedule, there were none who showed an improvement of minus 8s, 84, 83 

21, in the per cent of problems correctly solved; 3 showed an improvement of minus 20, 19, 18 
17, or 16; 1 showed minus IS, 14, 13. 12, or 11; etc. 



Time 
Dist. 


No. of 
Pupils 


Grade 


Ma- 
terial 


-85 
81 


-80 
76 


-75 
7i 


-70 
66 


-65 
61 


-60 
56 


-55 
5i 


-50 
46 


-45 
4i 


-40 
36 


-35 
3i 


-30 

26 

I 

9 
2 


-25 
21 


Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 


56 
58 

30 
31 

34 
35 

56 
7i 

175 
153 


7 
7 

6 
6 

6 
6 

■ 5 
5 

3>4 
3.4 


Geog. 
Geog. 

Hist. 
Hist. 

Geog. 
Geog. 

Div. 
Div. 

Add. 
Add. 


1 






1 


I 


3 


1 
1 


2 


1 

1 
1 


1 
1 

6 

5 


6 
I 


1 
1 

1 

7 
6 



(Above Table continued) 



Time 
Dist. 


-20 
16 


-15 
11 


-10 
6 

10 

3 

4 
1 

2 
2 

3 
1 

19 
10 


-5 
1 

11 

5 

6 
8 

4 
5 

5 
9 

8 
13 




4 

14 
9 

4 

7 

14 

7 

18 
32 

19 

17 


5 
9 

6 
9 

7 
9 

6 

5 

12 
9 

14 
17 


10 
14 

4 
13 

2 

4 

3 
9 

11 

4 

8 
18 


15 
19 

5 
5 

1 

2 

2 

2 

3 

12 
11 


20 
24 

1 
9 

2 

1 

3 

10 
8 


25 
29 

I 

3 

1 

1 

2 

7 
7 


30 
34 

1 

3 

8 


35 
39 

1 
1 

5 
4 


40 
44 

2 
3 


45 
49 

1 

2 


50 
54 

4 
4 


55 
59 

1 

1 


60 
64 

3 


65 
69 

1 


70 

74 

1 
3 


75 
79 


Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 


3 
1 

1 

2 
1 

15 

4 


1 
1 

1 

2 
2 

12 
3 


I 



The Lyndhurst Experiments with Grade Pupils 29 

COMPARISON OF THE TWO SORTS OF DISTRIBUTION 

(a) Geography and history. Next in order, after the discussion of 
the learning of new bonds from zero strength contained in Chapter I, 
we are able to compare the results obtained from the application of 
the two different schemes of distribution to the learning of geograph- 
ical and historical facts — bonds which had already been exercised 
to a considerable extent through the study of these subjects in the 
regular class work of the sixth and seventh grades. 

For the purpose of comparison, the seventh grade classes, whose 
time schedules are shown in Table III, on page 16 above, are divided 
into the Equal group, including classes 't' and V, which had sched- 
ules with practice-periods of equal length, and the Reducing group, 
including classes V, 'w', and V, which had practice-periods de- 
creasing in length and on the average much shorter than those of 
the Equal group. The experiment with these seventh grade classes 
extended over a period of twenty-six days, from the 5th to the 31st 
of March. The average length of the practice-periods for the Equal 
group was fifteen minutes, that for the Reducing group slightly less 
than eleven minutes. 

Also, we may divide the sixth grade classes into two similar groups. 
The time schedules for these classes are given in Table II, on page 16 
above. The Equal group included classes 'p' and 'r', which had 
schedules with practice-periods of twenty minutes in length. The 
Reducing group included classes 'q' and 's', which had schedules with 
practice-periods decreasing in length and averaging just twelve 
minutes. The experiment with these sixth grade classes extended 
over a period of fifteen days, from the 9th to the 24th of February. 

In both cases the results show a greater improvement for the 
groups which followed the Reducing schedule, roughly speaking, one- 
seventh more. The facts appear in Table IX. 



30 Improvement and the Distribution of Practice 

TABLE IX 

COMPARISON OF RESULTS OBTAINED FROM THE TWO SORTS OF DISTRIBUTION 
IN THE LYNDHURST EXPERIMENTS 

















Per cent 










Ma- 




Average 


Average 


in Init. 


Gain 




Grade 


No. 




No. 


Gross 


Test 


Time 


and 


of 


terial 


Average 


Correct 


Gain in 


Correct 


in 


Dist. 


Class 


Pupils 


Prac- 


Age 


in I nit. 


No. 


of Prob- 


Per cent 








ticed 




Test 


Correct 


lems At- 
tempted 


Correct 


Equal 


7, t, u 


56 


Geog. 


13 -52 


29.89 


40.84 


77-55 


1.44 


Reduc. 


7, v,w, 


















X 


58 


Geog. 


I2.8I 


30.67 


48.86 


74.88 


8.91 


Equal 


6 p 


34 


Geog. 


11.50 


45-50 


21.15 


76.74 


•47 


Reduc. 


6. q 


35 


Geog. 


12.37 


37-74 


24.29 


72.72 


1-54 


Equal 


6 r 


30 


Hist. 


12. 10 


31.10 


8.47 1 


72.67 


.87 


Reduc. 


6 s 


3i 


Hist. 


12.30 


39 -52 


40.25 


77.26 


3-42 



We have, then, to inquire whether other factors than the nature 
of the distribution of practice might have caused this difference. 

Teacher. The factor of the teacher was rendered constant so 
far as possible, by the facts (1) that both the initial and final tests, 
and all the daily practice, with few exceptions, were conducted by 
the author, and (2) that these exceptions came by pure chance, and 
thus would be as likely to favor one group as the other. 

Time of day. The factor of time of day was equalized for the 
two groups by experimenting with the classes in an order that was 
random, so far as the time of day was concerned. 

Amount of previous practice. This factor could in no wise be deter- 
mined, except in so far as being in the same grade in school might 
imply an equal amount of previous practice in any given function, 
as, for example, arithmetic, or geography. In comparing pupils of 
the same grade, taken at random as ours were, it would seem fair 
to suppose, for example, that thirty pupils (class c) in the fourth 

1 The very small gain shown by this class is probably due to special circumstances 
to be reported later. 



The Lyndhurst Experiments with Grade Pupils 31 

grade in one building would have had as much practice, let us say 
in the function of addition, as had thirty-one pupils (class f) in the 
fourth grade in another building in the same school system, in which 
the distribution of pupils among the several buildings was based 
upon the place of residence of the pupils. 

Age. Referring to Table IX, in which is given the average age for 
each group, it will be seen that there is a difference of less than half 
a year between the two groups in either of the three comparisons. 
On the whole, the two groups are substantially equal in point of age. 
For, if we combine the three Equal groups into one larger group, 
and the three Reducing groups into one larger group, we have an 
average age of 12.37 years for the Equal group and 12.49 years for 
the Reducing group. So small a difference as .12 of a year in the 
average age of two groups is practically negligible. 

Mental attitude of the pupils. Over this factor we could exercise 
very little control, owing to the short length of time in which we had 
charge of the classes. This factor, however, appeared to be fairly 
well distributed between the two groups, except in the case of a 
certain sixth grade class which learned history. It so happened that 
this particular class had been predetermined to follow the Equal 
schedule. On nearly every occasion when we entered this room the 
teacher was at swords' points, so to speak, with some of the larger 
boys in the class. This was noticeably so on the day of the final 
test, as a result of which the score of this class was materially 
reduced from what it otherwise might have been. Due allowance 
will be made for this disturbance at the proper time. 

Physical conditions. Such conditions as light, heat, ventilation, 
seating arrangement of the pupils, etc., were fairly uniform in all 
the buildings, since each building was in good repair and well kept. 
These advantages were shared equally by the two groups. 

Physiological conditions. During the time of our experiment there 
were no gross disturbances, such as epidemics, and no minor dis- 
turbances of any kind affecting one group more than the other. 

Interest and worry. The psychological factors of interest and 
worry, in so far as we were able to discern, were equally favorable 
to both groups. 



32 



Improvement and the Distribution of Practice 



Initial ability. The initial ability of the two groups differed some- 
what, but upon the whole was substantially the same. The facts 
appear as follows: 

AVERAGE INITIAL SCORES 



Subject 


Grade 


Equal Group 


Reducing Group 


Geography 
Geography 
History 


7 
6 
6 


29.89 (No. P. 56) 
45.50 (No. P. 34) 
31 .10 (No. P. 30) 


30.67 (No. P. 58) 
37.74 (No. P. 35) 
39.52 (No. P. 31) 



Since the correlation between initial score and gross gain, though 
probably positive, is low 1 the differences shown here would not 
seriously affect any substantial difference due to the distribution of 
practice. 

Differences in initial ability, however, may be eliminated by the 
'pairing off' method, i. e., leaving out the initially better from one 
group and the initially poorer from the other group, until the Equal 
and the Reducing groups consist of pupils of equal average initial 
ability. The following results, for the two groups in question, are 
obtained by this sort of treatment: 

comparison of results of the Equal and the Reducing GROUPS, 

AFTER EQUALIZING INITIAL ABILITY BY THE METHOD OF ' PAIRING OFF* 
THE INITIAL SCORES OF THE TWO GROUPS 



Group 


Grade 


Ma- 
terial 


No. of 
Pupils 


Initial Score 


Gross Gain 


Equal 
Reduc. 


7 
7 


Geog. 
Geog. 


52 (56) 

53 (58) 


30-77 
30.76 


(29.89) 
(30.67) 


41.96 (40.84) 
47.11 (48.86) 


Equal 
Reduc. 


6 
6 


Geog. 
Geog. 


28 (34) 
25 (35) 


43 04 
42.88 


(45-5o) 

(37-74) 


19.54 (21.15) 
29.24 (24.29) 


Equal 
Reduc. 


6 
6 


Hist. 
Hist. 


23 (30) 
22 (31) 


34.26 

34-18 


(3i-io) 
(39-52) 


9-I3 2 (8-47) 
38.77 (40.25) 



1 See especially the results of Chapman ('15) and Thorndike ('16). 

2 Special disturbances characterized this group. See the paragraph on Mental 
attitude of the pupils on page 3 1 . 



The Lyndhurst Experiments with Grade Pupils 33 

If the results thus obtained are compared with those given in 
Table IX, it will be seen that the equalization of initial ability does 
not reduce the advantage shown in favor of the Reducing group, but 
rather increases it. Combining the results, we have, as a net result 
of equalizing the initial ability of the two groups, a gain of 9.25 per 
cent on the initial score, for the Reducing groups, whereas the Equal 
groups suffer a loss of 1.98 per cent on their initial score by this 
method of treatment. Leaving out the doubtful case in which spec- 
ial disturbances occurred, the gross gain made by the Reducing 
groups is approximately one and a quarter times that made by the 
Equal groups. 

Recency of last practice before the final test. In the experiment with 
seventh grade pupils the Reducing group had a five-minute practice- 
period two days before the final test and a seven and one-half min- 
ute practice-period five days before the final test. The Equal group 
had a fifteen-minute practice-period five days before the final test. 
Thus, while the Reducing group had five minutes of practice two 
days before the final test, which the Equal group did not have, if we 
count back as far as five days before the final test, the Equal group 
had two and one-half minutes more practice in closing than had 
the Reducing group. 

This same sort of comparison holds true in the case of the sixth 
grade pupils. Here the Reducing group had five minutes of practice 
twenty-four hours before the final test, which the Equal group did 
not have. Within the last two days preceding the final test, how- 
ever, the Equal group had twenty minutes of practice, while the 
Reducing group had but twelve and one-half minutes. If the last 
five days before the final test are counted, the two groups each had 
an equal amount of practice, while, if we count in one more day, the 
Equal group had ten minutes more practice in the closing exercises 
than the Reducing group. 

Just how much influence may be due to such numerical differences 
in recency of the last practice before the final test cannot be satisfac- 
torily determined, for, as yet, the mathematics of the curve of for- 
getting is problematical, to say the least. 

Number of practice-periods. In the geography experiment with 
seventh grade pupils the Equal group had eight practice-periods in 
all, the Reducing group had eleven. In the geography and history 
experiments with sixth grade pupils the Equal groups had six 



34 Improvement and the Distribution of Practice 

practice-periods and the Reducing groups ten. The Reducing 
groups thus had approximately one-third more practice-periods 
than the Equal groups. This may be construed as an advantage in 
favor of the Reducing groups, since most investigations seem to show 
that any given amount of practice is more effective when distributed 
in a larger number of periods, providing, of course, that the periods 
are not made too short, i. e., unreasonably short. 

Interval between practice-periods. Assuming that both schedules 
are spread over the same total length of time, the interval between 
periods becomes a direct corollary of the number of periods, i. e., the 
more practice-periods there are, the shorter must necessarily be the 
average length of the intervals between. In our experiments, the 
time-interval for the Equal schedule varied from two to five days, 
with an average length of eighty-nine hours in the case of the 
seventh grade experiment and an average of seventy-two hours in 
the case of the sixth grade experiments. With the Reducing schedule 
the variation was from one to five days, with an average of sixty-two 
hours for the seventh grade experiment and forty hours for the sixth 
grade experiments. It should be noted in this connection that at 
least a three-day interval, over the week-end, is unavoidable in 
experiments with public school children. 

Summing up the experiments with geography and history, the 
Reducing schedule shows notably better results. No sure conclu- 
sions concerning the reasons for this superiority can be drawn, owing 
to the various factors entering in, many of which could not be very 
rigorously controlled. With such a large margin of variability in the 
results thus far considered, it would certainly be far from conserva- 
tive to attribute the advantage in favor of the Reducing schedule to 
the difference in the length of the practice-periods, or the intervals 
between practice-periods, alone. 

(b) Single-column addition. Classes 'a' to 'k' inclusive practised 
with single-column addition according to the time-schedule shown in 
Table IV. The bonds here exercised were, presumably, much older 
than those exercised in the case of geography and history and may, 
therefore, be considered as fairly well established. Assuming the 
learning curve to be parabolic in form, the older the bonds are, the 
less is the amount of improvement that may be expected from a 
given amount of practice : hence, if it were possible to compare the 
results obtained from the practice in geography with those obtained 



The Lyndhurst Experiments with Grade Pupils 



35 



from the practice in addition, we should expect the greater improve- 
ment to be shown in the case of the subject representing the newer 
bonds, viz., geography. 

In view of the fact that some of the above-mentioned classes con- 
tained pupils in both halves of the grade, it was decided to combine 
the results for the entire third and fourth grades. This procedure 
may be further justified because of the larger number of cases we 
are thus enabled to include in the two groups which are to be com- 
pared. We have, then, for the Equal group, classes 'e, f, g, h, i' and 
*j\ containing 56 third grade pupils and 119 fourth grade pupils, or 
a total of 175. The Reducing group includes classes 'a, b, c, d' and 
'k' and contains 94 third grade pupils and 59 fourth grade pupils, 
or a total of 153. The results of this experiment, which was carried 
out from November 2 to December 18, 1914, show approximately 
the same amount of gross gain for both groups. The facts appear 
in Table X. 

TABLE X 
comparison of results of Equal and Reducing groups 

IN THE CASE OF SINGLE-COLUMN ADDITION 



Time 
Dist. 


No. of 
Pupils 


Ave. 

Age 


Ave. No. 

Correct 

in Init. 

Test 


Ave. Gross 

Gain in 
No. Correct 


Per Cent in Init. 

Test Correct, 

of Problems 

Attempted 


Loss or 
Gain in 
Per Cent 
Correct 


Equal 
Reduc. 


175 

153 


9.22 
965 


12.69 
11.25 


6.07 
6.10 


5986 
52.92 


Loss 1.27 
Gain 6 . 94 



Factors conditioning improvement. Among the factors condition- 
ing improvement, the factors of the teacher, the time of day, physi- 
cal and physiological conditions, etc., need not be discussed again, 
since both the Equal and the Reducing groups were influenced alike 
in these respects. 

Age. The average age of the Reducing group is .43 of a year more 
than that of the Equal group, notwithstanding the fact that the 
Equal group contained a much larger per cent of fourth grade pupils. 
This fact, no doubt, constitutes an advantage in favor of the Equal 
group, since the lower the average age of pupils in any given grade, 
the brighter they are, as a rule. A further investigation in this 



36 Improvement and the Distribution of Practice 

particular case showed that the higher average age of the Reducing 
group was largely due to the presence of a number of 'repeaters' in 
one of the third grade classes belonging to this group. The follow- 
ing comparison of this class with the entire group of 150 third grade 
pupils, in respect to age-distribution, illustrates the point in ques- 
tion. 



Age 7 8 9 10 11 12 13 14 15 



Med. Ave. 
Age Age 



Class 'b' 





4 


9 


8 


6 


6, 


1 


1 


1 


10. 12 


10.36 


150 3rd Gr. 


12 


52 


38 


27 


10 


6 


3 


I 


1 


8.79 


9.00 



These facts, (1) that the Equal group contained sixty-eight per 
cent of fourth grade pupils, as against but thirty-nine per cent of 
fourth grade pupils in the Reducing group, and (2) that the average 
age of the Reducing group was practically a half year older than that 
of the Equal (this being due largely to the presence of retarded 
pupils in the Reducing group), both tended to favor the Equal group. 

Initial ability. The initial ability of the two groups was approxi- 
mately the same, the facts appearing as follows: 

AVERAGE INITIAL SCORES 

Subject Grade Equal Group Reducing Group 

Addition 3rd, 4th 12.69 (No. P. 175) 11.25 (No. P. 153) 

Assuming a slight positive correlation between initial scores and 
amount of improvement in any given function, the higher average 
score made by the Equal group in the initial test would mean a 
slight additional advantage for this group. If we eliminate this 
difference in initial scores by the method already adopted, viz., 
"The Pairing-off Method," we have the following results: 

COMPARISON OF RESULTS OF THE Equal AND THE Reducing GROUPS, AFTER 

EQUALIZING INITIAL ABILITY BY THE METHOD OF "PAIRING OFF" 

THE INITIAL SCORES OF THE TWO GROUPS 

Group No. Pupils Ave. Init. Score Ave. Gross Gain 

Equal 170, (175) 12.05 (12-69) 6.02 (6.07) 

Reduc. 143, (153) 12.02 (11.25) 6.25 (6.10) 

The above equalization of the two groups in respect to initial 
score does not affect the average age to any noticeable extent. 



The Lyndhnrst Experiments with Grade Pupils 37 

The result is to reduce the average age of the Equal group by but 
.01 of a year and to increase the average age of the Reducing group 
by the same trifle. 

Recency of the last practice before the final test. All of the classes 
composing the Equal group, except class 'i', had the 100 minutes of 
practice preceding the final test spread over the period of time from 
November 2 to November 24, on which date all of the pupils in both 
groups were given an intermediate test. After this date the Equal 
group had no further practice until the final test on December 18. 
Class T began on the day of the intermediate test and had its 100 
minutes of practice distributed between this date and the date of 
the final test. 

The classes composing the Reducing group had eighty minutes of 
practice spread over the period of time from November 2 to No- 
vember 24. Between this date and the final test all of these classes, 
except class 'k', were given the other twenty minutes of their prac- 
tice, the last two and one-half minutes of which occurred two days 
before the final test. Class 'k', because of a change in plan, was 
given no further practice until the final test, hence this class was 
short twenty minutes in the total amount of practice as compared 
with all the other classes taking part in the experiment. 

A comparison of the scores made in the intermediate test on 
November 24 with those made in the final test on December 18 
reveals marked differences between classes, in the amount of for- 
getting that took place during the interim between the two tests, in 
the case of those classes which had no practice during this time. 
Similar differences in respect to the amount of improvement that 
took place also appear between those classes which continued to 
practice during this time. Some of the classes which had no prac- 
tice during this interim made higher scores in the final test than in 
the intermediate test, while some of the classes which continued 
to practice during this interim made higher scores in the interme- 
diate test than in the final test. Differences in the amount of prac- 
tice done outside of the experiment, or differences in response to 
general school training may account, in part, for such discrepancies, 
but the probability is that most of these irregularities are due to 
chance variation. 



38 



Improvement and the Distribution of Practice 



(c) Short division. The very large per cent of improvement made 
by the classes in division was, no doubt, partly due to the novel ar- 
rangement of the examples. For any such novel arrangement of test 
material obviously operates to produce a lower initial score than 
otherwise might be made. The lower initial score, together with the 
rapid improvement, due to the ease of overcoming the novelty in the 
arrangement of the material, and the consequent higher score in the 
final test, all tend to produce a very great amount of improvement as 
a result of a relatively small amount of practice. Then, too, the 
practice curve for division in the case of our pupils was probably not 
as far advanced as the curve for addition. This would seem evident 
from the fact, (i) that pupils ordinarily learn addition before they 
learn division, and (2) that most pupils have occasion to use addition 
more than they do division. Hence, in so far as the curve of learn- 
ing is parabolic in form, the improvement would be more rapid in 
the case of that function which has been practised least — in this 
case, division. 

In conducting the experiment in short division the same general 
plan was followed as in the experiment with addition described 
above. The initial test was given on November 2, the intermediate 
test on November 24, and the final test was given on December 18, 
1914. Classes 'n' and V followed the Equal schedule and classes 
T and 'm' followed the Reducing schedule, as shown in Table IV. 
The Reducing group made approximately one-ninth more gain than 
the Equal. These facts appear in Table XI. 



TABLE XI 

comparison of results of the Equal and Reducing GROUPS, 

IN THE CASE OF SHORT DIVISION 



Time 
Dist. 


No. of 
Pupils 


Average 
Age 


Ave. No. 
Correct in 
Init. Test 


Ave. Gross 

Gain in 
No. Correct 


Per Cent in 

Init. Test 

Correct, 

of Problems 

Attempted 


Gain in 
Per Cent 
Correct 


Equal 
Reduc. 


56 
71 


H-45 
11.23 


65.98 
70.04 


64.66 
71.42 


87-54 
85.00 


6-57 
7.86 



The Lyndhurst Experiments with Grade Pupils 39 

Having already discussed the common factors influencing im- 
provement in our experiment, only those need be treated again 
whose variation is peculiar to the groups now being compared. 

Age. In this comparison the average age of the Equal group is 
.22 of a year higher than that for the Reducing group. This differ- 
ence was partly due to the fact that the classes composing the 
Equal group contained a larger per cent of pupils in the upper half 
of the grade. This may be considered as a slight advantage in favor 
of the Equal group, since the higher average age of this group was 
not due to the presence of retarded pupils as was the case in the 
comparison of the two groups made up of third and fourth grade 
pupils, (cf. p. 35 above.) 

Initial ability. The initial average score made by the Reducing 
group was approximately six per cent higher than that made by the 
Equal group. Assuming a slight positive correlation between initial 
score and improvement, this would constitute a slight advantage in 
favor of the Reducing group. If this assumption is true, then, by 
equalizing the initial ability of the two groups by the method em- 
ployed in previous comparisons, we should expect the greater gain 
shown for the Reducing group to be somewhat reduced. This is 
what happens, for by eliminating the higher scores made by the 
Reducing group, thus equalizing the average scores, for the two 
groups, the average gross gain for the Reducing group is reduced 
from 71.42 to 69.40. Thus the gain of the Reducing over the Equal 
group is reduced, approximately, from one-tenth to one- twentieth. 
Again, if we equalize the initial ability of the two groups by elimi- 
nating the highest two scores from the Reducing group and the low- 
est scores from the Equal group, we find that the average gross gain 
for the Reducing group is reduced from 71.42 to 69.88, while that 
for the Equal group is increased from 64.66 to 68.18. This reduces 
the margin shown in favor of the Reducing group to 1.7, or approxi- 
mately two and one-half per cent of the average gross gain. 

Recency of last practice preceding the final test. The Equal group 
had no practice from the intermediate test on November 24 to the 
final test on December 18. The Reducing group had twenty minutes 
of practice distributed over the interim between these two dates, 
with two and one-half minutes of this amount occurring two days 
before the final test. On the basis of general theory, therefore, this 
arrangement would be calculated to favor the Reducing group. A 



40 Improvement and the Distribution of Practice 

comparison of the records made in the intermediate test with those 
made in the final test, however, seems to indicate that the Equal 
group did not suffer because of this lack of recency of practice. Such 
discrepancies between expectation based on general theory, and 
results obtained in experiments of this kind, may be due to differ- 
ences in the amount of outside practice done by the two groups, or 
other factors which were beyond our control. In all probability, 
however, these and many other conflicting results occurring in simi- 
lar experiments are due as much to chance variation as to any other 
single factor. Such irregularities can only be smoothed out by the 
use of a large number of subjects, running well into the thousands. 
Summing up all the Lyndhurst experiments, the results seem to 
point toward a slight superiority in the arrangement of the Reducing 
schedule, though this superiority is by no means marked. The 
following section contains an account of the repetition of the same 
experiment with school children in another state. 



SECTION III 
THE OHIO EXPERIMENTS 

I. DESCRIPTION 

The experimental work reported in this section was done during 
the fall of 1915, in the following seven villages of Northwest Ohio, 
viz., Rocky Ridge, Lakeside, Elmore, and Oak Harbor, in Ottawa 
County; Greenwich, in Huron County; Waterville, in Lucas 
County, and Weston, in Wood County. These were all typical 
rural villages for this section of the country, with the exception of 
Lakeside, which is a summer resort. The school population of this 
last-named place, however, is made up of the permanent residents 
of the village. 

CHARACTER OF THE SCHOOL POPULATION 

The nationality of the pupils in the public schools of all the seven 
villages taken together would average about seventy-five per cent 
American born. Because of the lack of general educational oppor- 
tunity, however, they would not, upon the whole, rank above the 
Lyndhurst pupils in the use of the English language. In respect to 
economic conditions, there appeared to be greater extremes of both 
wealth and poverty than was noted at Lyndhurst. The general 
educational conditions, including school plant and equipment, aver- 
aged about the same, though with poorer buildings in some of the 
Ohio villages than any found in Lyndhurst or Rutherford. In 
respect to supervision and teaching force the comparison was very 
favorable. 

SUBJECTS USED IN THE EXPERIMENT 

The subjects used in the Ohio experiments consisted of all the 
pupils in grades three to eight, inclusive. Table XII shows the 
distribution of pupils by villages and by grades. 



42 Improvement and the Distribution of Practice 

TABLE XII 

DISTRIBUTION OF THE PUPILS USED IN THE OHIO EXPERIMENTS, 
BY VILLAGES AND BY GRADES 









Grades 








Villages 












Total 
















3 


4 


5 • 


6 


7 


8 




Rocky Ridge 


10 


7 


13 


9 


7 


11 


57 


Elmore 


22 


20 


15 


22 






79 


Lakeside 


9 


13 


15 


15 


10 


17 


79 


Greenwich 


27 


11 


13 


23 


12 


11 


97 


Waterville 


25 


17 


13 


23 


21 


16 


115 


Weston 


33 


20 


20 


21 


29 


18 


141 


Oak Harbor 


37 


(44) 


33 


28 


32 


43 


217 


Totals 


163 


132 


122 


141 


in 


116 


785 



The absence of any data for the seventh and eighth grades of the 
Elmore school is due to a misunderstanding on the part of the 
teacher of these grades with reference to the handling of the initial 
test papers, resulting in the loss of these records. In order to pro- 
vide a comparison of Ohio pupils with Lyndhurst pupils, in division, 
the forty-four fourth grade pupils of the Oak Harbor school were 
practised with short division. With this exception, all the third and 
fourth grade pupils were practised with adding, the fifth and eighth 
grades learned geographical facts, and the sixth and seventh grades 
learned historical facts. 

About 1,000 pupils in all entered into the experiment, but owing 
to illness, bad weather, and other causes, a number of the records 
were incomplete. After making all necessary eliminations, there 
were 785 records that could be used in our study. 

CONTENT-MATERIAL USED 

The material used in the Ohio experiments was identical with 
that used at Lyndhurst, viz., geography, history, addition, and 
short division. For a description of this material the reader is 
referred to a corresponding paragraph in Section II. 

PLAN OF THE EXPERIMENT 

In planning the time schedule, the same general principle was 
followed as in previous experiments, viz., to compare the effect of 



The Ohio Experiments with Grade Pupils 43 

two sorts of distribution, designated as the Equal and the Reducing. 
In this case, the Equal schedule consisted of eight practice-periods 
of fifteen minutes each (except the last but one, which was ten 
minutes) occurring, in so far as possible, on Mondays, Wednesdays, 
and Fridays. The Reducing schedule consisted of eleven practice- 
periods, distributed over approximately the same period of time, 
and apportioned as follows: 15, 15, 15, 12^, 10, 10, 7>^, 7^, 5, 
2}4 and the final test of 15 minutes. By this arrangement, each of 
the two schedules has a total of 115 minutes. The initial and final 
scores were counted as the average, or mid-point, of the first and 
last fifteen minutes of practice, respectively, hence the amount of 
practice that was measured was 100 minutes — the same as in the 
Lyndhurst experiments. 

Owing to the impossibility of reaching more than one of the vil- 
lages on the same day, it was necessary to begin the experiment in 
each village on a different day, also, to give the final tests on differ- 
ent days. (There was one exception to this, since it was possible to 
reach two of the places, viz., Rocky Ridge and Lakeside, during 
school hours of the same day.) 

Table XIII gives the exact dates on which the work was done in 
each of the seven villages. The general plan was to have the final 
tests occur several days before the beginning of the Christmas 
holidays. 



44 



Improvement and the Distribution of Practice 



TABLE XIII 

DATES ON WHICH THE EXPERIMENT WAS CARRIED OUT 
IN EACH OF THE SEVEN VILLAGES 

The figures in the body of the table represent the length of each practice-period in minutes. 
All the classes at Rocky Ridge, Lakeside, and Greenwich, together with grades 6 and 8 from Oak 
Harbor, followed the Equal schedule. Grades 3,5, and 7 from Oak Harbor and all the classes at 
Elmore, Waterville, and Weston followed the Reducing schedule. 



EgwaZ-schedule Classes 


Reducing 


-schedule Classes 




1915 Nov. 


Rocky 


Lake- 


Green- 


Oak Harbor 


El- Wa 


ter- 


Wes- 


Ridge 


side 


wich 




more vi 


lle 


ton 


T 16 










15 






W 17 








15 15 








T 18 


15 


15 




15 


15 






F 19 






15 


15 15 


15 






S 20 
















S 21 
















M 22 


15 


15 


15 


15 12K 


I2# 






T 23 








10 


10 






W 24 


15 


15 


15 


15 10 


10 






T 25 
















F 26 
















S 27 
















S 28 
















M 29 


15 


15 


15 


15 7K 


lA 1 


5 


15 


T 30 










lA 1 


5 


15 


W 1 


15 


15 


15 


15 7A 


5 1 


5 


15 


T 2 












*A 


I2>£ 


F 3 


15 


15 


15 


10 5 


2^ I 





10 


S 4 
















S 5 
















M 6 


10 


10 


10 


2y 2 


15 I 





10 


T 7 












lA 


VA 


W 8 








15 15 




lA 


VA 


T 9 


15 


15 








5 




F 10 






15 






2^ 




S 11 
















S 12 
















M 13 










I 


5 




1916 l Jan. 
















T 6 














5 


F 7 














*A 


S 8 
















S 9 
















M 10 














15 



1 On account of a fear of an epidemic of scarlet fever, the school at Weston was closed before 
the experiment was completed. The completion practice and the final test for this school, there- 
fore, were given as soon as school reconvened in January. 



The Ohio Experiments with Grade Pupils 45 

ADMINISTRATION OF THE EXPERIMENT 

All the initial and final tests were conducted by the author at 
the time most convenient for him. The daily practice, however, 
was in charge of the regular teachers and was done at the time of 
day most convenient for them. Thus, in the arrangement of the 
time schedules for the two groups, the factor of the time of day was 
distributed at random. 

Method of conducting the practice. In order that all the practice 
might be conducted as uniformly as possible, each teacher was 
provided with a set of instructions. The author was able to visit 
each village once or twice a week and thus checked up the work of 
both teacher and pupils. The instructions placed in the hands of 
the teachers were simple and very brief. First of all was a copy of 
the time schedule which was to be followed. Then came the follow- 
ing directions: "The teacher will simply follow the example set by 
the experimenter in giving the initial test. N.B. All the time 
allotted to each practice-period shall be used in practice, in the case 
of addition, or in study and testing, in the case of geography and his- 
tory. No individual help is to be given any of the pupils. In gen- 
eral, no questions are to be answered. The idea is for each pupil to 
strive to improve upon his, or her, own record made at the previous 
time. The pupils are not to see or handle the papers, except during 
the regular practice-periods, as provided for in the time schedule." 

Instead of suggesting difficulties by attempting to explain them 
beforehand, such problems were left to be dealt with whenever they 
might arise. Few such difficulties arose, however, and such as did 
arise were disposed of without any undue interruption of the main 
purpose of the experiment. 

The incentives used were the same as those used in the Lynd- 
hurst experiments, with the exception that no material rewards 
were offered. The teachers were instructed to conduct the prac- 
tice in their accustomed manner, so far as incentives were con- 
cerned. In a word, it was our purpose to have the practice in these 
experiments done as nearly as possible under the same conditions 
that prevail in the regular work of the school year. That this result 
was actually achieved, is borne out by the fact that among the 
individual records of the pupils were found not only marks of im- 
provement, but also the usual per cent of failures to profit by the 
practice. 



4 6 



Improvement and the Distribution of Practice 



SCORING AND TABULATING THE RESULTS 

The same method of scoring was used as in the Lyndhurst experi- 
ments. The daily scores made in addition and, in so far as possible, 
in geography and history, were kept by the teachers. The super- 
intendents, with but two exceptions, did the tabulating of the daily 
scores. The tabulating for the Greenwich school was done by 
Mr. Craig, a teacher in a nearby consolidated rural school. In 
the case of the Oak Harbor school, the tabulating was done by 
Miss Ingle, one of the teachers in the high school of that village. 
The author is responsible for checking the work done by the teach- 
ers and superintendents and for the statistical treatment of the data. 

2. RESULTS OF THE EXPERIMENTS 

Table XIV gives the distribution of scores made in the initial 
test. Table XV gives the distribution of the per cent of problems 
attempted that were solved correctly. Table XVI gives the dis- 
tribution of the changes that took place in the number of problems 
solved correctly, during the 100 minutes of practice. Table XVII 
gives the distribution of the changes in the per cent of accuracy. 



TABLE XIV 

INITIAL ABILITY IN THE OHIO EXPERIMENTS: 
FREQUENCIES FOR INITIAL SCORE 

The figures at the top represent the number of problems solved correctly in the initial test of 
IS minutes. 

The table reads as follows: Of the 39 eighth grade pupils who learned geographical facts according 
to the Equal schedule, there were none who made scores of o, plus 1, 2, 3, or 4; 1 made a score of 
5, 6, 7, 8, or 9; 10 made scores of 10, 11, 12, 13, 14; etc. 



Time 
Dist. 


No. 

of 

Pupils 


Grade 


Ma- 
terials 
Used 




4 

10 

12 

3 
1 

8 
1 

4i 
44 


5 
9 

1 
3 
11 
7 
13 
17 
16 
25 

23 

38 


10 

14 

10 
3 

15 
9 

17 
28 

21 
16 
29 
23 


15 
19 

7 
11 

18 
1 

H 

24 

11 

5 

12 

17 


20 
24 

6 
15 
11 

6 

12 
6 
1 

3 
9 


25 
29 

7 
10 

2 

5 

8 
6 

4 
3 


30 

34 

4 
8 

4 
5 

1 

2 

2 


35 
39 

4 
6 

2 

2 
2 

1 


40 
44 

4 
1 

1 


45 
49 

4 

1 
1 


50 

54 

1 
2 


55 
59 

1 
2 


60 

64 


Equal 

Reduc. 

Equal 

Reduc. 

Equal 

Reduc. 

Equal 

Reduc. 

Equal 

Reduc. 


39 
77 
61 
50 
47 
94 

74 
48 

114 

137 


8 
8 

7 
7 
6 
6 

5 

5 

3,4 

3,4 


Geog. 

Geog. 

Hist. 

Hist. 

Hist. 

Hist. 

Geog. 

Geog. 

Add. 

Add. 


1 



The Ohio Experiments with Grade Pupils 



47 



TABLE XV 



INITIAL ABILITY IN THE OHIO EXPERIMENTS: FREQUENCIES FOR 
PER CENT OF ACCURACY 

The figures at the top represent the per cent of problems attempted that were correctly solved in 
the initial test. 

The table reads as follows: Of the 39 eighth grade pupils who learned geographical facts according 

to the Equal schedule, there were none who showed as low a per cent of accuracy as o, 1, 2 14 

per cent; 1 fell as low as 15, 16, 17, 18, or 19 per cent; but no others fell within the range of 20, 
21, 22 29 per cent; 3 showed an accuracy of 30, 3r, 32, 33, or 34 per cent; etc. 



Time 


No. of 


Grade 


Materials 





5 


10 


15 


20 


25 


30 


35 


40 


Dist. 


Pupils 


Used 


4 


9 


H 


19 


24 


29 


34 


39 


44 


Equal 


39 


8 


Geog. 




1 






3 




1 


Reduc. 


77 


8 


Geog. 


2 




I 




2 


1 


1 




2 


Equal 


61 


7 


Hist. 




1 


1 


1 






1 


2 




Reduc. 


50 


7 


Hist. 








1 


3 




3 


1 


4 


Equal 


47 


6 


Hist. 










1 




2 


2 




Reduc. 


94 


6 


Hist. 






I 




1 




2 


3 


3 


Equal 


74 


5 


Geog. 


1 




3 


2 


1 


2 


3 


4 


6 


Reduc. 


48 


5 


Geog. 








3 


4 


5 


7 


4 


1 


Equal 


114 


3.4 


Add. 


12 


2 


4 


4 


4 


5 


6 


3 


9 


Reduc. 


137 


3,4 


Add. 


13 


3 


5 


7 


5 


9 


5 


9 


10 













(A bove 


Table 


continued) 










Time 


45 


50 


55 


60 


65 


70 


75 


80 


85 


90 


95 


100 


Dist. 


49 


54 


59 


64 


69 


74 


79 


84 


89 


94 


99 


Equal 


4 


5 


4 


4 


4 


3 


3 


2 


2 


2 


1 


1 


Reduc. 


4 


6 


1 


13 


5 


10 


n 


8 


5 


4 




1 


Equal 


1 


2 




1 


2 


3 


8 


7 


8 


13 


6 


4 


Reduc. 


3 


3 


5 


2 


3 


6 


3 


2 


3 


2 


1 


5 


Equal 


2 


1 


2 


10 


6 


4 


5 


5 


1 


5 




r 


Reduc. 


5 


5 


3 


11 


10 


8 


8 


14 


4 


8 


2 


6 


Equal 


1 


4 


8 


6 


5 


7 


10 


4 


3 


2 


I 


1 


Reduc. 


2 


5 


3 


5 


1 


4 


2 




1 






1 


Equal 


3 


14 


4 


3 


8 


9 


3 


2 


7 


4- 




5 


Reduc. 


8 


12 


4 


5 


5 


9 


14 


3 


2 


7 


1 


1 



4 8 



Improvement and the Distribution of Practice 



TABLE XVI 

AMOUNT OF IMPROVEMENT IN THE OHIO EXPERIMENTS: 
NUMBER OF PROBLEMS SOLVED CORRECTLY 

The figures at the top represent the amount of improvement, plus or minus, that took place during 
the time of the experiment, in the number of problems solved correctly in 15 minutes. 

The table reads as follows: Of the 39 eighth grade pupils who learned geographical facts according 

to the Equal schedule, there were none who showed an improvement of minus 20, 19, 18 1; 

none showed an improvement of 0, plus 1, 2, 3 9; 1 made an improvement of io, 11, 12, 13, 

or 14; 3 made an improvement of 15, 16, 17, 18, or 19; etc. 



25 
29 



Time 
Dist. 


No. of 
Pupils 


Grade 


Equal 


39 


8 


Reduc. 


77 


8 


Equal 


61 


7 


Reduc. 


50 


7 


Equal 


47 


6 


Reduc. 


94 


6 


Equal 


74 


5 


Reduc. 


48 


5 


Equal 


114 


3,4 


Reduc. 


137 


3,4 



Materials 


-20 


-15 


-10 


-5 





5 


10 


15 


20 


Used 


16 


11 


6 


1 


4 


9 


14 


19 


24 


Geog. 














1 


3 


2 


Geog. 














2 


4 


11 


Hist. 


1 




2 






1 




7 


6 


Hist. 












1 


4 


5 


4 


Hist. 








1 


1 


3 


3 


5 


8 


Hist. 










1 




5 


3 


12 


Geog. 






I 


5 


10 


5 


4 


6 


6 


Geog. 










1 




4 


5 


7 


Add. 




3 


12 


23 


39 


21 


8 


2 


5 


Add. 




2 


6 


23 


43 


27 


12 


9 


3 



{Above Table continued) 



Time 


30 


35 


40 


45 


50 


55 


60 


65 


70 


75 


80 


85 


90 


Dist. 


34 


39 


44 


49 


54 


59 


64 


69 


74 


79 


84 


89 


up 


Equal 


3 




4 


7 


7 


4 






2 


2 


1 


1 


1 


R.educ. 


6 


3 


7 


5 


10 


5 


7 


3 


3 


1 


3 


1 


3 


Equal 


8 


8 


5 


4 


7 


2 


2 




2 










Reduc. 


4 


4 


12 


3 


3 


1 






1 










Equal 


4 


4 


5 


3 


2 


1 
















Reduc. 


14 


17 


17 


9 


7 


2 


1 














Equal 


7 


7 


7 


3 


5 
















1 


Reduc. 


10 


5 


6 


2 




















Equal 




i 
























Reduc. 


4 


1 




2 








2 













The Ohio Experiments with Grade Pupils 



49 



TABLE XVII 



AMOUNT OF IMPROVEMENT IN THE OHIO EXPERIMENTS: PER CENT CORRECT 

The figures at the top represent the amount of improvement, plus or minus, in the per cent of 
problems attempted that was correctly solved, that took place during the time of the experiment. 

The table reads as follows: Of the 39 eighth grade pupils who learned geographical facts according 
to the Equal schedule, there were none who showed an improvement in accuracy of minus 85, 84, 

83 16 per cent; 1 showed an improvement of minus is, 14, 13, 12, or n per cent; none 

showed minus 10, 9, 8, 7, or 6 per cent; 1 showed minus 5, 4, 3, 2, or 1 per cent; 3 showed o, plus 
1, 2, 3, or 4 per cent; etc. 



Time 
Dist. 


No. of 
Pu- 
pils 


Grade 


Ma- 
terial 


Less 
than 
-46 


-45 
4i 


-40 
36 


-35 
3i 


-30 
26 


-25 
21 


— 20 
16 


-15 
11 


— 10 
6 


-5 

1 


Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 

Equal 
Reduc. 


39 

77 

61 
50 

47 
94 

74 
48 

114 

137 


8 
8 

7 
7 

6 
6 

5 
5 

3.4 

3.4 


Geog. 
Geog. 

Hist. 
Hist. 

Hist. 
Hist. 

Geog. 
Geog. 

Add. 
Add. 


2 

1 

2 
3 


1 

1 

1 

2 
2 


1 

5 
1 


2 

4 
3 


1 

8 
6 


2 
2 

1 
1 

3 
6 


1 

2 
1 

1 

8 

4 


1 

2 
1 

1 

6 
11 


2 

3 

5 

3 
1 

7 
8 


1 

1 
1 

1 
8 

11 
11 



{Above Table continued) 



Time 





5 


10 


15 


20 


25 


30 


35 


40 


45 


50 


55 


60 


65 


70 


Dist. 


4 


9 


14 


19 


24 


29 


34 


39 


44 


49 


54 


59 


64 


69 


up 


Equal 


3 


6 


2 


4 


3 


1 


7 


3 


5 




3 










Reduc. 


2 


7 


12 


8 


13 


7 


6 


4 


5 


3 


2 


2 




1 


3 


Equal 


2 


11 


4 


6 


6 


4 


8 


4 


2 


1 


2 










Reduc. 


5 


3 


6 




5 


5 


2 


3 


4 


4 


2 


4 


3 


1 


2 


Equal 


5 


2 


4 


2 


7 


3 


8 


2 


2 


2 






2 






Reduc. 


8 


8 


10 


13 


4 


10 


9 


9 


4 


6 


I 


2 


2 






Equal 


6 


6 


7 


9 


6 


6 


4 


4 


1 




5 


1 






1 


Reduc. 


1 


1 


2 


2 


3 


6 


3 


4 


9 


6 


6 




1 




2 


Equal 


8 


4 


7 


7 


9 


2 


11 


3 


1 


1 




3 


1 




1 


Reduc. 


7 


11 


11 


5 


4 


10 


9 


6 


7 


1 


2 


4 


2 


1 


2 



50 Improvement and the Distribution of Practice 

The Equal and the Reducing groups were made up from the 
pupils of the seven villages as follows : The Equal group included all 
the classes at Rocky Ridge, Lakeside, and Greenwich, and grades 
3, 5, and 7 from Oak Harbor. The Reducing group included grades 
6 and 8 from Oak Harbor and all the classes at Elmore, Waterville, 
and Weston. 

The third and fourth grades (except the fourth grade at Oak 
Harbor) practiced with single-column addition. The fifth and 
eighth grades learned geographical facts, and the sixth and seventh 
grades learned historical facts. The material used and the amount 
of practice measured were the same as in the case of the Lyndhurst 
experiments. 

The results of the geography and history experiments were influ- 
enced in the same way, though not to so great an extent, as at 
Lyndhurst, i. e., by a little preliminary study before the initial test. 
Thus there was in the initial test a slight immediate memory ele- 
ment, which tended to make the initial score higher than it other- 
wise would have been. In no case, however, was there any practice 
immediately preceding the final test. This precluded any immedi- 
ate memory effect on the final score and operated to make the 
amount of gain shown more conservative. It should be understood 
in this connection that the influence here mentioned does not in any 
way affect the comparison of the two schedules of distribution of 
practice, but only affects the amount of improvement shown as a 
result of the ioo minutes of practice. 

The results of these experiments show that in the case of the 
third, fourth, fifth, and sixth grades, the Reducing groups made 
more improvement. In the case of the seventh and eighth grades 
the Equal groups did better. Table XVIII gives the complete facts. 



The Ohio Experiments with Grade Pupils 
TABLE XVIII 

COMPARISON OF THE RESULTS OF THE Equal AND THE Reducing GROUPS, 
GRADE BY GRADE 



51 



Time 


No. 
of 


Grade 


Ma- 
terial 


Average 


Ave. No' 
Correct 


Ave. 

Gross 
Gain in 


Per Cent 
in Init. Test 
Correct of 


Gain 
in 
Per 


Dist. 


Pupils 




Used 


Age 


in Init. 
Test 


No. 
Correct 


Problems 
Attempted 


Cent 
Correct 


Equal 


39 


8 


Geog. 


13-69 


21.65 


53 02 


65 -77 


29.38 


Reduc. 


77 


8 


Geog. 


13 


64 


24 


25 


48.57 


65.20 


25-85 


Equal 


61 


7 


Hist. 


12 


93 


16 


52 


34 48 


65.67 


12.56 


Reduc. 


50 


7 


Hist. 


12 


78 


17 


34 


32.70 


62.98 


3268 


Equal 


47 


6 


Hist. 


12 


09 


11 


66 


28.07 


67-I5 


16.27 


Reduc. 


94 


6 


Hist. 


11 


77 


15 


91 


35-22 


69-55 


22.00 


Equal 


74 


5 


Geog. 


10 


96 


14 


74 


24.90 


57-93 


14.14 


Reduc. 


48 


5 


Geog. 


11 


09 


9 


96 


28.63 


45-81 


34-82 


Equal 


114 


3.4 


Add. 


8 


92 


8 


90 


5.70 


43.28 


5-90 


Reduc. 


137 


3.4 


Add. 


9 


07 


9 


47 


7.07 


45 85 


8-95 



FACTORS CONDITIONING IMPROVEMENT 

The factor of the teacher was equalized by a random selection of 
the classes which made up the two groups. 

The hour of the day at which the practice was done was deter- 
mined by the individual teachers, and was thus as likely to favor 
one group as the other. It should be stated, however, that, having 
selected a suitable hour of the day, each teacher maintained this 
hour throughout the experiment. 

It is not at all certain that the pupils in the different villages had 
had the same amount of previous training, grade for grade, in the 
subjects used. Yet, the random method of selecting the pupils 
would tend to favor one group as much as the other, in so far as this 
factor was concerned. 

No exceptional cases of mental attitude were noticed in any of 
the schools, which might have unduly influenced either group. 

The physical conditions, such as light, heat, ventilation, seating 
arrangement of the pupils, etc., differed very greatly among the 



52 Improvement and the Distribution of Practice 

several schools. Each group, however, contained some of the poor- 
est and some of the best conditions of this sort. The school building 
at Oak Harbor was the only modern building found, hence some of 
the classes in this school were included in each of the two groups. 
(cf. Table XIII.) 

Conditions of health influenced the work of the pupils in practi- 
cally all of the schools. The time of the experiments happened to 
be an exceptionally bad season for epidemics of scarlet fever, 
measles, whooping cough, and the like. The records of pupils thus 
affected, however, were necessarily eliminated on account of in- 
completeness, except in the case of the Weston school, which was 
closed as a precautionary measure against the scarlet fever about a 
week before our experiment was concluded. The pupils of this 
school returned, however, in good condition immediately after the 
holidays, at which time the experiment was again taken up and 
finished. 

The factors of interest and worry varied from week to week and 
with the different schools. The influence of such factors cannot be 
measured or rated in any satisfactory way, but, upon the whole, we 
should say, the pupils in all the schools manifested the ordinary 
amount of interest in the tasks assigned them and appeared as care- 
free as any ordinary school children. 

There is a very close correspondence between the two groups in 
respect to average age, in all the comparisons made. The largest 
difference in any instance is that of the sixth grade, in which the 
Equal group shows an average age of .32 of a year more than the 
Reducing group. This difference may be due, in part at least, to the 
slower progress made by the pupils in the schools at Rocky Ridge 
and Lakeside, which two schools happened to fall in the Equal 
group. The slower progress of the pupils of these schools may be 
accounted for by the fact that many of them are kept out of school 
for a considerable time during the fall season, in order to help har- 
vest the apple crop. It will be noticed that in the case of the 
seventh and eighth grades, also, the Equal groups are slightly older 
than the Reducing groups. 

In four out of the five comparisons the Reducing groups show 
slightly higher initial scores than the Equal groups, but, on the 
whole, the two groups are substantially equal in this respect. The 
facts appear as follows: 



The Ohio Experiments with Grade Pupils 



53 



AVERAGE INITIAL SCORES 



Subject 


Grade 


Equal Group 


Reducing Group 


Geography 

History 

History 

Geography 

Addition 


8 

7 
6 

5 
3-4 


21.65 (No. P. 39) 
16.52 (No. P. 61) 

11.66 (No. P. 47) 
14.74 (No. P. 74) 

8.90 (No. P. 114) 


24.25 (No. P. 77) 

17.34 (No. P. 50) 

15.91 (No. P. 94) 

9.96 (No. P. 48) 

9.47 (No. P. 137) 



By the application of the "pairing off" method, described on a pre- 
vious page, for the equalization of the initial ability of the two 
groups, we obtain the following results: 

comparison of results of the Equal and the Reducing GROUPS, 

AFTER EQUALIZING INITIAL ABILITY BY THE METHOD OF "PAIRING 
OFF" THE INITIAL SCORES OF THE TWO GROUPS 



Group 


Grade 


Ma- 
terial 


No. of Pupils 

39 (39) 
61 (77) 


Initial Score 


Gross Gain 


Equal 
Reduc. 


8 
8 


Geog. 
Geog. 


21.65 (21.65) 
21.75 (24-25) 


53-02 (53 
5023 (48 


02) 
57) 


Equal 
Reduc. 


7 
7 


Hist. 
Hist. 


61 (61) 
49 (50) 


16.52 (16.52) 
16.51 (17.34) 


34-48 (34 
32.78 (32 


48) 
70) 


Equal 
Reduc. 


6 
6 


Hist. 
Hist. 


44 (47) 
70 (94) 


12.07 (n.66) 
12.44 (i5-9i) 


28.86 (28 
35-6o (35 


07) 
22) 


Equal 
Reduc. 


5 
5 


Geog. 
Geog. 


57 (74) 
48 (48) 


10.25 (14-74) 
9.96 (9.96) 


21-35 (24 
28.63 (28 


90) 
63) 


Equal 
Reduc. 


3.4 
3,4 


Add. 
Add. 


114 (114) 
135 (137) 


8.90 (8.90) 
8.87 (9.47) 


5-7o (5-70) 
6-34 (7-07) 



If the results thus obtained are compared with those given in 
Table XVIII, it will be seen that the equalization of initial ability, 
upon the whole, increases the advantage shown in favor of the 
Reducing groups. Combining the results, we have, as a net result 
of equalizing the initial ability of the two groups, a gain of 7.71 per 
cent on the initial score for the Equal group and a gain of 23.06 
per cent on the initial score for the Reducing group. All in all, the 
gross gain made by the Reducing groups is about one-fourteenth 
more than that made by the Equal groups. 

In respect to recency of last practice before the final test, the two 
groups were about equally favored. 



54 



Improvement and the Distribution of Practice 



The Equal groups had eight practice-periods and the Reducing 
groups had eleven, spread over approximately .the same total 
length of time. The average interval between periods would, 
therefore, necessarily be shorter in the case of the Reducing groups. 
Compare, in this connection, the paragraph on "Plan of the Experi- 
ment," and the discussion on "Number of Practice-periods," above. 

Summarizing the data contained in Table XVIII, we have the 
following comparison as a basis for predicting what might be 
expected from an extended series of similar experiments. 



SUMMARY OF AVERAGE GROSS GAIN IN THE NUMBER OF PROBLEMS 
CORRECTLY SOLVED, OR WORDS CORRECTLY SUPPLIED 



Material 


Grade 


Equal Groups 


Reducing Group 


Reducing/ Equal 


Geography 


8 


53-02 


48.57 


92 


History 


7 


34-48 


32.70 


95 


History 


6 


28.07 


35-22 


125 


Geography 


5 


24.90 


28.63 


1. 15 


Addition 


3 and 4 


5-70 


7.07 


1 .24 



From the above summary we have, as an average, Reducing j Equal, 
1. 10 with a P. E. t. — obt. Av. of .042. There are thus about five 
chances in a hundred that the results from an extended series of 
similar experiments would show no advantage for the Reducing 
group. The chances are about four to one that the ratio will be 
between 1.02 and 1.18. Or, we may say, the chances are even that 
the ratio will be between 1.06 and 1.14; or, again, the chances are 
about ninety-nine to one that the ratio will not exceed the limits 
of .975 and 1.225. 

This difference in favor of the Reducing schedule of practice is 
probably due, in part, to the larger number of practice-periods in 
this schedule; and, in part, to the more intensive application on the 
part of the pupils in their effort to get as much done as possible, 
since the periods gradually become shorter and shorter as the 
experiment nears completion. The knowledge of a more limited 
time in which to work at any given task, at each succeeding period, 
may serve as a stimulus to bring about increased speed in 'warming 
up' and greater concentration throughout the period. 



SECTION IV 

THE EXPERIMENT WITH NORMAL SCHOOL STUDENTS 

The experimental work reported in this section was done during 
the second semester of the school year 191 5-19 1 6, with first year 
students in the State Normal College of Bowling Green, Ohio, act- 
ing as subjects. 

THE NATURE OF THE EXPERIMENT 

The problem was to compare the results obtained from a given 
amount of practice (in this case, sixty minutes), when distributed 
according to two different schedules, previously described and 
designated as the Equal and the Reducing. 

The material used was the Thorndike single-column addition 
sheets, which are described on page 14 above. The subjects were 
157 first year students in the Normal College, of which number all 
were women except fourteen. The general plan of the experiment 
was the same as in all our previous experiments, viz., to divide the 
subjects into two groups and practice one group according to the 
Equal schedule and the other according to the Reducing schedule. 
It was also possible in this case to group the subjects according 
to their initial ability, as determined by the record made in the first 
ten minutes of practice. 

There were 170 students who entered the experiment, but only 
157 of the records were complete enough to count. The arrange- 
ments of the time schedule, together with the exact dates on which 
the practice was done, are shown in Table XIX. 

Administration of the experiment. The students met in four regu- 
lar class sections on Mondays, Wednesdays, and Fridays at 8:00, 
9:00 and 10:00 o'clock a. m., and 2:00 o'clock p. m., respectively. 
The initial test was given at the regular class session of each section 
on Friday, March 10. The papers were scored and then sorted into 
four groups in the following manner: The papers were first arranged 



56 Improvement and the Distribution of Practice 

TABLE XIX 

TIME SCHEDULE FOLLOWED IN THE EXPERIMENT 
WITH NORMAL SCHOOL STUDENTS 

The figures in the body of the table represent the length of the practice-periods. 



1916 


Equal Group 


Reducing Group 


Sec. A 


Sec. B 


Sec. A 


Sec. B 


F Mar. 10 


10 


10 


10 


10 


S Mar. 11 










S Mar. 12 










M Mar. 13 


10 


5 


15 


2^ 


T Mar. 14 




5 






W Mar. 15 




5 


10 


5 


T Mar. 16 


10 


5 






F Mar. 17 




5 


VA 


lV* 


S Mar. 18 










■S Mar. 19 










M Mar. 20 


10 


5 


5 


10 


T Mar. 21 




5 






W Mar. 22 


10 


5 


2^ 


15 


T Mar. 23 










F Mar. 24 


10 


10 


IO 


10 



in order, according to the magnitude of the scores made. The paper 
with the highest score was placed in pile No. 1, the next highest in 
pile No. 2, the next in pile No. 3, the next in pile No. 4, the next in 
pile No. 1, and so on. The four piles of papers thus represented four 
groups of students with approximately equal initial ability. The 
papers in pile No. 1 were marked "Equal Group, Section A," those 
in pile No. 2 were' marked "Reducing Group, Section A," those in 
pile No. 3 were marked "Equal Group, Section .B," and those in 
pile No. 4 were marked "Reducing Group, Section B." The papers 
were returned to the students on the following day, whereupon each 
person could see what his initial score was and also to which prac- 
tice group he had been assigned. According to previous instructions 
each person then proceeded to carry out the practice in adding, 
according to the time schedule for the group to which he had been 
assigned. At each regular recitation period the progress of the 
experiment was checked up by the author, who had charge of the 
course which these students were taking. 



The Experiment with Normal School Students 



57 



The initial and final tests were given in class, the averages being 
taken as the initial and final scores. The daily practice was done 
outside the class. The students were furnished with correct answers 
to the problems and were thus enabled to check up their own work 
after each practice-period, without receiving extra practice in adding 
by so doing. Each one was required to plot a graph of his work, 
showing the improvement made as a result of the sixty minutes of 
practice. A score of one was allowed for each column correctly 
added. The index of accuracy used was: 

Number of columns correct 



Number of columns attempted 



THE RESULTS OF THE EXPERIMENT 



Table XX gives the frequencies for the number of problems 
solved correctly in the initial test of ten minutes. Table XXI gives 
the frequencies for the per cent of problems attempted that were 
correctly solved in the initial test. Table XXII gives the frequencies 
for the amount of improvement made as a result of the fifty minutes 
of practice that was measured, i. e., that amount of practice which 
lay between the average, or mid-point, of the first ten minutes of 
practice and the average, or mid-point, of the last ten minutes of 
practice. 

TABLE XX 



NORMAL SCHOOL EXPERIMENT: FREQUENCIES FOR INITIAL SCORE 

The figures at the top represent the number of problems solved correctly in the initial test of 10 
minutes. 

The table reads as follows: Of the 42 subjects, who practised adding according to Equal schedule 
A, there were none who made scores as low as io, 11, 12, 13, or 14 in the initial test; 3 made 
scores of 15, 16, 17, 18, or 19; 7 made scores of 20, 21, 22, 23, or 24; etc. 



Group 


No. 


10 


15 


20 


25 


30 


35 


40 


45 


50 


55 


60 


and Section 


Subjects 


14 


19 


24 


29 


34 


39 


44 


49 


54 


59 


64 


Equal — A 


42 




3 


7 


8 


8 


6 


4 


3 


2 


1 




Equal — B 


4i 


2 


3 


6 


10 


9 


4 


4 


2 


1 






Reduc— A 


36 




2 


5 


9 


10 


4 


3 


2 




1 




Reduc— B 


38 


1 


3 


7 


9 


5 


5 


3 


2 


1 


1 


1 


Equal A and B 


83 


2 


6 


13 


18 


17 


10 


8 


5 


3 


1 




Reduc. AandB 


74 


1 


5 


12 


18 


15 


9 


6 


4 


1 


2 


1 



58 



Improvement and the Distribution of Practice 



TABLE XXI 

NORMAL SCHOOL EXPERIMENT: FREQUENCIES FOR ACCURACY 
The figures at the top represent the per cent of problems attempted that were correctly solved in 
the initial test of 10 minutes. 

The table reads as follows: Of the 42 subjects who practised adding according to Equal schedule 
A, there was 1 who fell as low as 40, 41, 42, 43, or 44 per cent in accuracy; none fell within the 

range of 45, 46 69 per cent; 3 showed 70, 71, 72, 73, or 74 per cent; none showed 75, 76, 

77, 78, or 79 per cent; 8 showed an accuracy of 8o, 81, 82, 83, or 84 per cent; etc. 



Group 
and Section 


No. 
Subjects 


40 
44 


45 
49 


50 
54 


55 
59 


60 

64 


65 
69 


70 
74 


75 
79 


80 

84 


85 
89 


90 
94 


95 
99 


100 


Equal — A 
Equal — B 

Reduc— A 
Reduc— B 

Equal A and B 
Reduc. A and B 


42 
41 

36 
38 

83 
74 


1 

1 










3 

2 

3 
2 


3 
1 

1 
1 

4 
2 


1 

4 
5 

1 

9 


8 

5 

8 
6 

13 
14 


8 
6 

6 
7- 

H 
13 


10 
10 

7 
9 

20 
16 


7 
10 

8 
6 

17 


5 
5 

2 
2 

10 

4 



TABLE XXII 

NORMAL SCHOOL EXPERIMENT: FREQUENCIES FOR IMPROVEMENT IN 
THE NUMBER OF PROBLEMS SOLVED CORRECTLY 

The figures at the top represent the amount of improvement, plus or minus, that took place during 
the time of the experiment, in the number of problems solved correctly in 10 minutes. 

The table reads as follows: Of the 42 subjects who practised adding according to Equal schedule 
A, there was 1 who showed an improvement of minus 10, 9, 8, 7, or 6; 1 who showed an improve- 
ment of minus 5, 4, 3, 2, or 1; 8 who showed an improvement of o, plus 1, 2, 3, or 4; etc. 



Group 


No. 


10 


5 





5 


10 


15 


20 


25 


30 


and Section 


Subjects 


6 


1 


4 


9 


14 


19 


24 


29 


34 


Equal — A 


42 


I 


1 


8 


12 


9 


9 


1 


1 




Equal — B 


4i 






12 


12 


10 


4 


2 




1 


Reduc— A 


36 






10 


9 


7 


5 


4 




I 


Reduc— B 


38 




2 


7 


8 


8 


12 


1 






Equal A and B 


83 


1 


1 


20 


24 


19 


13 


3 


1 


1 


Reduc. A and B 


74 




2 


17 


17 


15 


17 


5 




1 



DISCUSSION OF THE RESULTS 

In this experiment both sections of the Reducing grojup made 
more improvement than the corresponding sections of the Equal 



The Experiment with Normal School Students 



59 



group. Combining the results of the two sections of each group and 
comparing the totals thus obtained, the Reducing group shows 
approximately one-fourteenth more gain than the Equal group. 
The complete facts appear in Table XXIII. 

TABLE XXIII 
comparison of results of the Equal and the Reducing GROUPS, 

IN THE EXPERIMENT WITH NORMAL SCHOOL STUDENTS 



Group 


No. 

of 

Subjects 


Average 
Age 


Ave. No. of 
Problems 
Correct in 
Init. Test 


Ave. Gross 
Gain in 
No. of 

Problems 
Correct 


Per Cent 

Correct 

in 

Init. Test 


Gain 

in 

Per Cent 

Correct 


Equal — A 


42 


19.22 


32.76 


9-74 


88.40 


5.00 


Reduc. — A 


36 


20.36 


31-44 


10.62 


88.11 


5-56 


Equal — B 


41 


20.04 


29.88 


9-36 


89.46 


3-13 


Reduc— B 


38 


19.61 


31.46 


9.92 


86.50 


6.32 


Equal— A, B 


83 


19-63 


3i-3i 


9-55 


88.93 


4.06 


Reduc— A, B 


74 


19.98 


3144 


10.27 


87.30 


5-94 



From the above table we have the following summary of results 
for average gross gain in the number of problems correctly added: 



Sections 


Equal Group 


Reducing Group 


Reducing/ Equal 


A 

B 

A, B 


9-74 
9-36 
9-55 


10.62 

9.92 

10.27 


1 .09 
1 .06 
1-075 



From the two comparisons of the A and B sections of the two 
groups, we have, as an average, Reducing I Equal, 1.075, which has a 
P. E. /. — obt. Av. of .0073. There is thus about one chance in a 
hundred thousand or so that the results from an extended series of 
similar experiments would show no advantage for the Reducing 
group. The chances are about ninety-nine to one that the ratio will 
be between 1.046 and 1.104. 



6o 



Improvement and the Distribution of Practice 



Performing a similar operation with the data for the gain in per 
cent of problems correct, we have the following results: 



Sections 


Equal Group 


Reducing Group 


Reducing/ Equal 


A 

B 

A, B 


5.00 

313 

4.07 


5-56 
6.32 

5-94 


1 .10 
2.02 
1-47 



Comparing the A and B sections of the two groups, we have, as 
an average, Reducing! Equal, 1.56, which has a P. E. /. — obt. Av. of 
.022. This means that in an extended series of similar experiments 
the Reducing group would practically always show the greater gain 
in the per cent of problems correct. 

Taking as a basis of comparison, the median gross gain in the 
number of problems correct (c/. Table XXIII.), fifty-six per cent of 
the Reducing group equal or exceed the median improvement made 
by the Equal group. 

The factors conditioning improvement in this experiment were, in 
general, the same as those conditioning the improvement made by 
school children in the two larger experiments. So far as affecting 
one group more than the other is concerned, none of these factors 
need be discussed here, except the following: 

The length of the practice-periods. Section A of the Equal group 
had, in all, six practice-periods of 10 minutes each. Section B of 
this group had an initial test of 10 minutes, then 8 periods of 5 min- 
utes each, with a 10-minute period at the close. Section A of the 
Reducing group had an initial and final practice of 10 minutes each, 
with the intervening 40 minutes distributed in 5 periods as follows : 
x 5) I0 » 7/^> 5> an d 2^2 minutes. The B section of this group had 
the same number of periods and of corresponding lengths, but the 
order of occurrence was increasing instead of decreasing, i. e., as 
follows: initial practice of 10 minutes, then 2}4, 5, 7/^, 10, and 15 
minutes, with a final practice of 10 minutes. 

According to the results the A section of each group made more 
gain than the B section of the same group. In each case, however, 
the sections of the Reducing group made more gain than the corre- 
sponding sections of the Equal group, roughly speaking, about one- 
tenth more. 



The Experiment with Normal School Students 61 

The number of practice-periods. The A section of the Equal group 
had one less practice-period than the A section of the Reducing 
group. The B section of the Equal group had three more practice- 
periods than had the B section of the Reducing group. Roughly 
speaking, the average length of the practice-periods for the Equal 
group was 7^ minutes, that for the Reducing group 8^ minutes. 
If the Reducing group suffered by the reverse arrangement of the 
schedule for the B section, the Equal group appeared to suffer also 
by the five-minute periods of its B section. 

In all other respects, such as the total length of time covered, the 
time-interval between periods, recency of last practice before the 
final test, etc., both groups were equally favored. 



SECTION V 
SUMMARY OF RESULTS OF ALL THE EXPERIMENTS 

The essential facts concerning initial score and gross gain in the 
number of problems solved correctly, for all our experiments, are 
summarized in Table XXIV. Since, in general, the Equal and 
Reducing groups are about equal in initial ability, we may utilize the 
results shown in this table for a rough measure of the relative 
advantage of the two schedules as follows: 

Using the last three columns, we have, as an average Reducing I 
Equal, 1.096, which has a P. E. /. — obt. Av. of .022. There are thus 
about two chances in a thousand that the results from an extended 
series of similar experiments would show no advantage for the 
Reducing schedule. The chances are even that the ratio will range 
between 1.07 and 1.12. Or, we may say, the chances are about four 
to one that the ratio will lie between 1.05 and 1.14; or, about nine 
to one that it will lie between 1.03 and 1.16; or, again we may say, 
the chances are about ninety-nine to one that in an extended series 
of similar experiments the ratio will fall between 1.01 and 1.18, and 
in only two cases out of a thousand would the ratio be 1.00, or less 
(i. e., in favor of the Equal schedule). 

If, however, instead of using the rough data given in Table XXIV, 
we use that obtained by equalizing the two groups in respect to 
initial ability, we have slightly different results. These data are 
given in Table XXV. From this table we obtain as an average 
Reducing I Equal, 1.125. The P. E. /. — obt. Av. of this average is 
.033. There are thus about ten chances in a thousand that the 
results from an extended series of similar experiments would show 
no advantage for the Reducing group. The chances are even that 
the ratio will be between 1.09 and 1.16 or, we may say, the chances 
are about ninety-six to four that it will not exceed the limits of 
1.03 and 1.22. 

If the Equal and Reducing groups are compared in respect to the 
gain in per cent of problems correct, it is found that the latter group 
surely did better. The facts appear in Table XXVI. From the last 



Summary of Results of All the Experiments 



63 



three columns of this table we have as an average Reducing /Equal, 
2.64, which yields a P. E. /. — obt. Av. of .371. There are thus about 
three chances out of a thousand that in an extended series of similar 
experiments the Reducing schedule would show no advantage. The 
possible significance of the results of a similar treatment based on 
corrected data, after equalizing the two groups in respect to initial 
ability, would hardly warrant the labor of making the calculation. 
It could not notably alter the general results obtained from 
Table XXVI. 

TABLE XXIV 



SUMMARY OF AVERAGE INITIAL SCORE AND AVERAGE GROSS GAIN IN NUMBER OF 

problems solved correctly by the Equal and Reducing GROUPS, 

IN ALL THE EXPERIMENTS 







Material 


Initial Score 


Gross Gain 


Reduc.f 


Place 


Subjects 










Equal 














Used 


Eq. Gp. 


Re. Gp. 


Eq. Gp. 


Re. Gp. 


(Gain) 


T. C. 


Adults 


French 


00.00 


00.00 


110.00 


1 1 1 . 70 


1 .02 


Lynd. 


7th Gr. 


Geog. 


29.89 


30.67 


40.84 


48.86 


1 .20 


Lynd. 


6th Gr. 


Geog. 


45 -50 


37-74 


21.15 


24.29 


115 


Lynd. 


6th Gr. 


Hist. 


31.10 


39 -52 


8.47 1 


40.25 


(4-75?) 


Lynd. 


3rd, 4th Gr. 


Add. 


12.69 


11.25 


6.07 


6. 10 


1 .00 


Lynd. 


5th Gr. 


Div. 


65.98 


70.04 


64.66 


71.42 


1 . 10 


Ohio 


8th Gr. 


Geog. 


21.65 


24.25 


53 02 


48.57 


.92 


Ohio 


7th Gr. 


Hist. 


16.52 


17-34 


34-48 


32.70 


•95 


Ohio 


6th Gr. 


Hist. 


11.66 


15-91 


28.07 


35-22 


125 


Lynd. 


5th Gr. 


Geog. 


14-74 


9.96 


24.90 


28.63 


115 


Ohio 


3rd, 4th Gr. 


Add. 


8.90 


9-47 


5-7« 


7.07 


1 .24 


B.G.N.C. 


Adults 


Add. 


3I-3I 


31-44 


9-55 


10.27 


1.08 



Ave. Reducing/ Equal 1 . 096 

S. D. dis 108 

P. E. t. —obt.Av 022 

1 On account of special disturbances influencing this group, the data are omitted 
from the final calculations. 



6 4 



Improvement and the Distribution of Practice 



TABLE XXV 

SUMMARY OF AVERAGE INITIAL SCORE AND AVERAGE GROSS GAIN IN NUMBER OF 

PROBLEMS SOLVED CORRECTLY, AFTER EQUALIZING THE INITIAL SCORES OF THE 

TWO GROUPS BY THE METHOD OF "PAIRING OFF" THE SCORES MADE 

BY ONE GROUP WITH THOSE MADE BY THE OTHER GROUP 



Place 


Subjects 


Material 
Used 


Initial Score 


Gross Gain 


Reduc./ 
Equal 
(Gain) 


Eq. Gp. 


Re. Gp. 


Eq. Gp. 


Re. Gp. 


T. C. 


Adults 


French 


00.00 


00.00 


110.00 


in .70 


1.02 


Lynd. 


7th Gr. 


Geog. 


30-77 


30.76 


4i 


96 


47.11 


1.12 


Lynd. 


6th Gr. 


Geog. 


43-04 


42.88 


19 


54 


29.24 


1.50 


Lynd. 


6th Gr. 


Hist. 


34.26 


34-18 


9 


13 2 


38.77 


(4.24?) 


Lynd. 


3rd, 4th Gr. 


Add. 


12.05 


12.02 


6 


02 


6.25 


1.04 


Lynd. 


5th Gr. 


Div. 


68.01 


67-73 


68 


18 


69.88 


1.03 


Ohio 


8th Gr. 


Geog. 


21.65 


21-75 


53 


02 


50.23 


•95 


Ohio 


7th Gr. 


Hist. 


16.52 


16.51 


34 


48 


32.78 


•95 


Ohio 


6th Gr. 


Hist. 


12.07 


12.44 


28 


86 


35-6o 


1.23 


Ohio 


5th Gr. 


Geog. 


10.25 


9.96 


21 


35 


28.63 


i-34 


Ohio 


3rd, 4th Gr. 


Add. 


8.90 


8.87 


5 


70 


6-34 


1 . 11 


B.G.N.C. 


Adults 


Add. 


3i.3i 


31-44 


9 


55 


10.27 


1.08 



Ave. Reducing/ Equal 1 . 125 



S. D. dis. . 164 



P. E. t.—obt.Av. .033 



TABLE XXVI 

SUMMARY OF FACTS CONCERNING AVERAGE GAIN IN ACCURACY, 
FOR BOTH GROUPS IN ALL THE EXPERIMENTS 



Place 


Subjects 


Material 
Used 


Initial per Cent 
Correct 


Gain in per Cent 
Correct 


Reduc./ 
Equal 




Eq. Gp. 


Re. Gp. 


Eq. Gp. 


Re. Gp. 


(Gain) 


Lynd. 


7th Gr. 


Geog. 


77-55 


74.88 


1.44 


8.91 


6. 19 


Lynd. 


6th Gr. 


Geog. 


76 


74 


72.72 


•47 


1 


54 


3.28 


Lynd. 


6th Gr. 


Hist. 


72 


67 


77.26 


.87 2 


3 


42 


(3-93?) 


Lynd. 


3rd, 4th Gr. 


Add. 


59 


86 


52.92 


— I .27 3 


6 


94 


5.46? 


Lynd. 


5th Gr. 


Div. 


87 


54 


85.00 


6-57 


7 


86 


1 .20 


Ohio 


8th Gr. 


Geog. 


65 


77 


65.20 


29.38 


25 


85 


.88 


Ohio 


7th Gr. 


Hist. 


65 


67 


62.98 


12.56 


32 


68 


2.60 


Ohio 


6th Gr. 


Hist. 


67 


15 


69-55 


16.27 


22 


00 


i-35 


Ohio' 


5th Gr. 


Geog. 


57 


93 


45.81 


14.14 


34 


82 


2.46 


Ohio 


3rd, 4th Gr. 


Add. 


43 


28 


45-85 


5-9o 


8 


95 


1-52 


B.G.N.C. 


Adults 


Add. 


88 


93 


87.30 


4.06 


5 


94 


1 .46 



P. E. t.—obt. Av. .371 
group, the data are omitted 



Ave. Reducing/ 'Equal 2.64 S. D. dis. 1.74 

2 On account of special disturbances influencing this 
from the final calculations. 

3 Our conclusions are only rendered more conservative by treating this negative gain 
as though it were positive. 



Summary of Results of all the Experiments 65 

The final result is thus an advantage of five to fifteen per cent in 
favor of practice distinguished chiefly by decreasing period-lengths 
and increasing intervals, but also by other differences. Until the 
effect of these other factors has been more fully studied the psy- 
chology of the results obtained here remains somewhat obscure. For 
educational practice the outstanding fact is that the same amount 
of time gave notably better results when distributed according to 
the Reducing schedule than when distributed evenly. 



CHAPTER VI 

THE AMOUNT OF IMPROVEMENT AND ITS RELATION 
TO INITIAL ABILITY 

THE AMOUNT OF GROSS GAIN 

In respect to amount of gain made, our results, in general, sup- 
port the conclusions already drawn from similar experiments by 
Thorndike, Kirby, Chapman, and others, in that a very great 
amount of gain is shown from a relatively small amount of practice. 
Comparatively speaking, however, the amount of gain made by our 
subjects was considerably less than that reported by the above- 
named authors. The following figures show the comparison be- 
tween certain of the results published by Dr. Kirby and certain of 
our own results. 



ROUGH COMPARISON OF AMOUNT OF GAIN MADE BY SOME OF OUR OWN SUBJECTS 
AND THE SUBJECTS USED BY KIRBY 



Investigator 


No. 
of 


Grade 


Ma- 
terial 


Amount of 

Practice 

Measured 


Ave. No. of 

Problems 
Solved Cor- 
rectly per 
Min. in the 
Init. Test 


Ave. Gross 

Gain in No. 

of Problems 

Solved 




Pupils 




Used 


in Terms of 
Minutes 


Correctly 
per Minute 


Kirby 


732 


4th 


Add. 


60 


i-7 


•7 


Cummins 


266 


4th 


Add. 


100 


1 .0 


• 5 


Kirby 


606 


3rd, 4th 


Div. 


50 


3-4 


2.8 


Cummins 


171 


4th, 5th 


Div. 


IOO 


3-i 


2.9 



The amount of gain here shown is approximately 50 to 100 per 
cent on the initial score. That which is least expected, however, is 
that our subjects, working under ordinary school conditions similar 
to those reported by Kirby, and using identical material, made less 
gain in 100 minutes of practice than his pupils made in 60 minutes. 



Improvement and Its Relation to Initial Ability 



67 



Of course, no one would expect that any group of pupils would keep 
up so rapid a rate of improvement for any great length of time, 
probably not for, say, 1,000 minutes, and possibly not for 500 
minutes. Our own results tend to show, however, that the limit of 
rapid improvement had not yet been reached with 100 minutes of 
practice. The figures below give, for certain of our subjects, the 
rate of improvement for the first 80 minutes of practice and the 
rate for the concluding 35 minutes, on the basis of the number of 
problems solved in 15 minutes. 









Ave. No. of 


Ave. No. of 


Ave. No. of 


Number 




Ma- 


Problems 


Problems 


Problems 


of 


Grade 


terial 


Correct in 


Correct in 


Correct in 


Pupils 






Initial 
Test 


Intermediate 
Test 


Final Test 


153 


3. 4 


Add. 


10.25 


15-73 


17-35 


71 


5 


Div. 


70.04 


102.92 


141 .46 



From the above figures we have, in the case of the third and fourth 
grade pupils, an average gain, in the number of problems solved 
correctly in fifteen minutes, of .34 of a problem for every five min- 
utes of practice from the initial test to the intermediate test, i. e., 
the first eighty minutes of practice. The average gain for the next 
thirty-five minutes, i. e., up to the final test, was .23 of a problem 
for every five minutes of practice. In the case of the fifth grade 
pupils, the results show a very decided increase for the last thirty- 
five minutes of practice. The rate of increase for every five min- 
utesof the first eighty minutes of practice was 2.06 problems, while 
for the remaining thirty-five minutes it was 5.51 problems for every 
five minutes spent in practice. 

In the case of our subjects who learned geography and history 
the amount of gain appears even greater than in the case of the 
subjects just reported upon. The great amount of gain made was 
probably due (1) to the fact that the bonds exercised were newer, 
(2) to the nature of the material used, and (3) to the method of 
scoring used, all of which have already been discussed in previous 
sections. 



68 



Improvement and the Distribution of Practice 



INITIAL SCORE AND GROSS GAIN 

The assumption set forth in previous sections, viz., that gross 
gain is positively correlated with initial score, is borne out by all 
our results. Tables XXVII, XXVIII, XXIX, and XXX exhibit 
the facts in a way that requires little comment. 

These tables are not really suitable for calculating correlation co- 
efficients, but in order to get a notion of the relation comparable with 
other relations, I have computed the correlations by the method of 
unlike-signed pairs, which gives the following coefficients: 



Material Used 


No. of Pupils 


Grades 


Correlation between 

Initial Score and 

Gross Gain 


Geography 
History 
Addition 
Division 


421 

312 

579 
171 


5, 6, 7, 8, 
6,7 
3,4 
4,5 


• 13 
.06 
.16 

.64 



By a further grouping of the frequencies contained in the four 
tables, we have the results in the form shown in Table XXXI. 
A comparison of the low-initial- score group in each case with the 
high-initial-score group shows roughly the amount of correlation 
between initial score and amount of gain. The exceedingly high 
correlation shown in the case of the fourth and fifth grade pupils, 
who practised short division, is a fact which deserves further inves- 
tigation. 



Improvement and Its Relation to Initial Ability 



69 



TABLE XXVII 

AMOUNT OF IMPROVEMENT IN RELATION TO INITIAL SCORE, IN THE CASE OF 42 1 5TH, 
6TH, 7TH, AND 8TH GRADE PUPILS WHO LEARNED GEOGRAPHICAL FACTS 
The table reads as follows: Of the 19 pupils with an initial score of o, 1, 2, 3, or 4, there were none 
who made an improvement of less thano; 5 made an improvement of from o to 9; 4 from 10 to 19; etc. 

Amount of Improvement in Number of Words Supplied 



Init. 


Less 





10 


20 


30 


40 


50 


60 


70 


80 




Score 


than 


9 


19 


29 


39 


49 


59 


69 


79 


up 




0- 4 




5 


4 


3 


2 


1 


2 


1 


1 




19 


5- 9 


3 


7 


6 


H 


7 


e 


1 




1 


I 


45 


10-14 


1 


5 


8 


11 


12 


12 


3 


2 






54 


15-19 


1 


1 


4 


7 


6 


6 


6 


2 


1 


I 


35 


20-24 


1 




4 


8 


8 


7 


14 


5 


2 


2 


5i 


25-29 


1 


4 


4 


8 


10 


10 


4 


4 


3 


6 


55 


30-34 




3 


7 


9 


6 


15 


6 


3 


3 


3 


55 


35-39 


1 


1 


3 


6 


10 


8 


6 




2 


2 


39 


40-44 




3 


4 


3 


5 


3 


4 


3 


I 


1 


27 


45-49 






4 


3 


5 




3 




1 


1 


17 


50-54 






3 


3 


5 




2 


1 




1 


15 


55-59 








2 


2 


1 










5 


60-up 




1 


1 


1 






1 








4 




8 


30 


52 


78 


78 


68 


5J 


21 


*5 


/<? 


421 



TABLE XXVIII 

AMOUNT OF IMPROVEMENT IN RELATION TO INITIAL SCORE, IN THE CASE OF 312 

6TH AND 7TH GRADE PUPILS WHO LEARNED HISTORICAL FACTS 

{This table reads the same as Table XXVII) 

Amount of Improvement in Number of Words Supplied 



Init. 


Less 





10 


20 


30 


40 


50 


60 




Score 


than 


9 


19 


29 


39 


49 


59 


up 




0- 4 


1 




4 


5 


2 


4 






16 


5- 9 




1 


9 


13 


7 


7 


6 


1 


44 


10-14 


2 


1 


8 


14 


13 


19 


8 


1 


66 


15-19 




6 


5 


10 


15 


19 


8 


1 


64 


20-24 


3 


1 


3 


4 


10 


7 


5 


2 


35 


25-29 


2 


4 


4 


1 


11 


6 






28 


30-34 


2 


3 


4 




9 


1 


2 


1 


. 22 


35-39 


1 




1 


1 


3 


5 




1 


12 


40-44 


1 


2 




1 


3 




3 




10 


45-49 






2 


1 


1 








4 


50-54 






1 






3 


1 


1 


6 


55-59 






1 




1 


1 






3 


60-up 












I 


1 




2 




12 


18 


42 


50 


75 


73 


34 


8 


312 



70 



Improvement and the Distribution of Practice 
TABLE XXIX 



AMOUNT OF IMPROVEMENT IN RELATION TO INITIAL SCORE, IN THE CASE OF 579 3RD 
AND 4TH GRADE PUPILS WHO PRACTISED SINGLE-COLUMN ADDITION 

{This table reads the same as Table XXVII) 
Amount of Improvement in the Number of Problems Added Correctly 



Init. 


Less 


— 10 


-5 





5 


10 


15 


20 


25 


30 


35 




Score 


than — 11 


-6 


— 1 


4 


9 


14 


19 


24 


29 


34 


up 




0- 4 






19 


69 


33 


8 


8 


6 




2 


1 


146 


5- 9 




3 


29 


30 


26 


25 


13 


6 


1 




1 


134 


10-14 




13 


22 


46 


22 


14 


9 


7 


3 




1 


137 


15-19 


4 


6 


19 


17 


14 


10 


5 


4 


3 


1 




#J 


20-24 


1 


6 


6 


10 


12 


3 


4 






3 


1 


46 


25-29 


1 


3 


3 


1 


2 


4 


1 




1 




1 


17 


30-34 


2 




2 


2 




2 




1 






1 


10 


35-up 










1 


2 




1 






2 







8 


31 


100 


175 


no 


<55 


40 


25 


8 


6 


8 


579 



TABLE XXX 

AMOUNT OF IMPROVEMENT IN RELATION TO INITIAL SCORE, IN THE CASE OF 171 4TH 
AND 5TH GRADE PUPILS WHO PRACTISED SHORT DIVISION 

(This table reads the same as Table XXVII) 
Amount of Improvement in Number of Problems Solved Correctly 



Init. 

Score 


Less 

than 





9 


10 
19 


20 
29 


30 
39 


40 
49 


50 
59 


60 
69 


70 
79 


80 
89 


90 
99 


100 
119 


120 
up 




0- 9 
10-19 
20-29 
30-39 
40-49 
50-59 
60-69 
70-79 
80-89 

90-99 
1 00-119 
120-139 
140-up 


1 
1 
1 

1 

1 

5 


4 
2 
1 

1 

8 


6 
1 
1 
I 
I 

I 
I 

12 


1 


2 
7 
3 
3 
1 
1 
2 

9 


1 


3 
1 

5 
3 

1 

3 
6 


1 
3 
3 
1 
2 
3 

2 
2 

17 


4 
1 

2 

3 
1 

3 
1 

2 

17 


1 
1 

2 
2 

3 
2 

2 
1 

1 

15 


1 

1 
2 
2 
6 

2 

3 
2 
1 

20 


1 

1 

3 
2 
1 

1 
9 


1 

1 

2 
2 

1 

1 

8 


1 

2 
1 

1 
2 

3 
2 

10 


1 

2 
1 
1 
1 

4 
3 
2 

*5 


J4 

17 
17 
15 
17 
18 
12 
12 

7 

12 

8 

_5_ 

171 



Improvement and Its Relation to Initial A bility 
TABLE XXXI 

CONDENSATION OF TABLES XXVII, XXVIII, XXIX, AND XXX 
WITH THE MEDIANS INSERTED FOR EACH GROUP 

5th, 6th, 7th, and 8th Grade Pupils (Geography) 



7i 





Less 





10 


20 


^0 


40 


SO 


60 


70 


80 








than 


9 


19 


29 


39 


49 


59 


69 


79 


up 


Total 


Median 


0-14 


4 


17 


18 


28 


21 


18 


6 


3 


2 


1 


118 


27.14 


15-24 


2 


1 


8 


15 


14 


13 


20 


7 


3 


3 


86 


42.31 


25-34 


1 


7 


11 


17 


16 


25 


11 


7 


6 


9 


no 


41.20 


35-up 


1 

8 


5 
30 


15 
52 


18 
78 


27 
78 


12 

68 


16 

53 


4 
21 


4 

15 


5 
I* 


107 
421 


35-37 







6th and 7th Grade Pupils (History) 





Less 
than 



9 


10 
10 


20 
29 


30 
39 


40 
49 


50 
59 


60 

up 


Total 


Median 


0- 9 


1 


1 


13 


18 


9 


11 


6 


1 


60 


28.33 


10-14 


2 


1 


8 


14 


13 


19 


8 


1 


66 


36.15 


15-24 


5 


11 


12 


15 


36 


32 


13 


3 


127 


35 69 


25-up 


4 
12 


5 
18 


9 

42 


3 
50 


17 

75 


11 

73 


7 
34 


3 
8 


59 

312 


32.06 







3rd and 4th Grade Pupils (Addition) 





Less 




























than 


— 10 


—5 





5 


10 


15 


20 


25 


30 


35 


Total 


Median 




— 11 


— b 


— I 


4 


9 


14 


19 


24 


29 


34 


up 






0- 4 






19 


69 


33 


8 


8 


6 




2 


1 


146 


3-91 


5" 9 




3 


29 


30 


26 


25 


13 


6 


1 




1 


134 


5-96 


10-14 




13 


22 


46 


22 


14 


9 


7 


3 




1 


137 


3.64 


15-up 


8 


15 


30 


30 


29 


21 


10 


6 


4 


4 


5 


162 


4.67 




8 


31 


700 


175 


no 


68 


40 


25 


8 


6 


8 


579 





72 



Improvement and the Distribution of Practice 
4th and 5th Grade Pupils (Division) 





Less 





10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


120 


Total 


Median 




than 


9 

7 


19 

8 


29 
12 


39 

4 


49 

4 


59 

4 


b 9 


79 


89 


99 


ny 


up 






0-29 


2 


4 


1 


1 


1 






48 


25-83 


30-49 


1 




2 


4 


8 


5 


1 


5 


3 


1 


1 


1 


I 


32 


42.00 


50-79 








3 


4 


4 


5 


4 


10 


6 


4 


3 


3 


47 


73-50 


80-up 


2 


1 


2 






4 


7 


2 


6 


1 


2 


6 


11 


44 


76.67 




5 


8 


12 


19 


id 


^ 


17 


'5 


20 


9 


5 


JO 


15 


171 





